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Properties

Label 1-3997-3997.279-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.956 + 0.291i$
Analytic cond. $429.537$
Root an. cond. $429.537$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.975 − 0.218i)2-s + (−0.528 − 0.849i)3-s + (0.904 − 0.426i)4-s + (−0.761 − 0.648i)5-s + (−0.701 − 0.712i)6-s + (0.789 − 0.614i)8-s + (−0.441 + 0.897i)9-s + (−0.884 − 0.466i)10-s + (−0.0385 + 0.999i)11-s + (−0.840 − 0.542i)12-s + (−0.391 + 0.920i)13-s + (−0.148 + 0.988i)15-s + (0.635 − 0.771i)16-s + (0.537 + 0.843i)17-s + (−0.234 + 0.972i)18-s + (−0.0715 + 0.997i)19-s + ⋯
L(s)  = 1  + (0.975 − 0.218i)2-s + (−0.528 − 0.849i)3-s + (0.904 − 0.426i)4-s + (−0.761 − 0.648i)5-s + (−0.701 − 0.712i)6-s + (0.789 − 0.614i)8-s + (−0.441 + 0.897i)9-s + (−0.884 − 0.466i)10-s + (−0.0385 + 0.999i)11-s + (−0.840 − 0.542i)12-s + (−0.391 + 0.920i)13-s + (−0.148 + 0.988i)15-s + (0.635 − 0.771i)16-s + (0.537 + 0.843i)17-s + (−0.234 + 0.972i)18-s + (−0.0715 + 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.956 + 0.291i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.956 + 0.291i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05885821866 - 0.3952410319i\)
\(L(\frac12)\) \(\approx\) \(-0.05885821866 - 0.3952410319i\)
\(L(1)\) \(\approx\) \(1.172881646 - 0.4069594169i\)
\(L(1)\) \(\approx\) \(1.172881646 - 0.4069594169i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (0.975 - 0.218i)T \)
3 \( 1 + (-0.528 - 0.849i)T \)
5 \( 1 + (-0.761 - 0.648i)T \)
11 \( 1 + (-0.0385 + 0.999i)T \)
13 \( 1 + (-0.391 + 0.920i)T \)
17 \( 1 + (0.537 + 0.843i)T \)
19 \( 1 + (-0.0715 + 0.997i)T \)
23 \( 1 + (-0.277 - 0.960i)T \)
29 \( 1 + (-0.191 + 0.981i)T \)
31 \( 1 + (0.986 + 0.164i)T \)
37 \( 1 + (-0.857 - 0.514i)T \)
41 \( 1 + (-0.851 + 0.523i)T \)
43 \( 1 + (-0.170 + 0.985i)T \)
47 \( 1 + (-0.350 - 0.936i)T \)
53 \( 1 + (0.509 - 0.860i)T \)
59 \( 1 + (0.401 + 0.915i)T \)
61 \( 1 + (-0.802 + 0.596i)T \)
67 \( 1 + (-0.999 - 0.0110i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.874 + 0.485i)T \)
79 \( 1 + (0.999 + 0.0220i)T \)
83 \( 1 + (-0.266 + 0.963i)T \)
89 \( 1 + (0.968 - 0.250i)T \)
97 \( 1 + (-0.685 - 0.728i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.83891669284428892561280147442, −17.69124742859685200791627519813, −17.21993689930838264880795660513, −16.36408632429220860238291706782, −15.72902565865024887627958674530, −15.37003853494196402712439064161, −14.84147197789693065282669633659, −13.84691288741802707976416962301, −13.51812891699153853782723117255, −12.22286997665538504822632088247, −11.84687979005647248820025924625, −11.259900499837304158324269273798, −10.61251421086476499270563222161, −10.01595278943187221711192045749, −8.921467393463998349640553319189, −7.984464950588249125805066965070, −7.406756688290524580825319559270, −6.50921359002363811746266978106, −5.88739931989090431209533342771, −5.11098286302650549508343939583, −4.57566854119102657087525651554, −3.49872093542981144483159354526, −3.27619577424943057686843234705, −2.47326929769459689359789832775, −0.80813965281785557885130431043, 0.050557784253267059451297848957, 1.384430482307532429397909322334, 1.67432944155887174172672724289, 2.707436408735638945003036505576, 3.78000846542603337528682847237, 4.46597290311443594487897877989, 5.059413133831858651936048577708, 5.82676608190403185598164241231, 6.71337960822238116688815644669, 7.161912156539184611645120518166, 7.97815312399956348612245047145, 8.64069047679645938384849101426, 9.95974220404896499344675895393, 10.48940734668655372307661416634, 11.45799790138628116178104886190, 12.08721124857384732673789868757, 12.32960427343323332655957800913, 12.927057213280882982019136275125, 13.69443947082119215776510027516, 14.618134834153571624703606752221, 14.903735493211337348817138030573, 16.03084121716584587026771603387, 16.576432331231972498040270853753, 16.96608871870137297775175804234, 18.04511814788484243292783344429

Graph of the $Z$-function along the critical line