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Properties

Label 2-403-403.25-c1-0-9
Degree $2$
Conductor $403$
Sign $0.301 - 0.953i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.216i·2-s + (1.35 + 2.35i)3-s + 1.95·4-s + (0.933 + 0.538i)5-s + (0.509 − 0.294i)6-s + (−2.50 + 1.44i)7-s − 0.856i·8-s + (−2.19 + 3.79i)9-s + (0.116 − 0.202i)10-s + (1.69 + 0.979i)11-s + (2.65 + 4.59i)12-s + (−3.20 − 1.65i)13-s + (0.313 + 0.542i)14-s + 2.92i·15-s + 3.72·16-s + (−2.22 − 3.85i)17-s + ⋯
L(s)  = 1  − 0.153i·2-s + (0.784 + 1.35i)3-s + 0.976·4-s + (0.417 + 0.240i)5-s + (0.208 − 0.120i)6-s + (−0.946 + 0.546i)7-s − 0.302i·8-s + (−0.731 + 1.26i)9-s + (0.0368 − 0.0638i)10-s + (0.511 + 0.295i)11-s + (0.766 + 1.32i)12-s + (−0.888 − 0.458i)13-s + (0.0836 + 0.144i)14-s + 0.756i·15-s + 0.930·16-s + (−0.539 − 0.934i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.301 - 0.953i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66891 + 1.22325i\)
\(L(\frac12)\) \(\approx\) \(1.66891 + 1.22325i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.20 + 1.65i)T \)
31 \( 1 + (-5.33 + 1.60i)T \)
good2 \( 1 + 0.216iT - 2T^{2} \)
3 \( 1 + (-1.35 - 2.35i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.933 - 0.538i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.50 - 1.44i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.69 - 0.979i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.22 + 3.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.53 + 2.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.663T + 23T^{2} \)
29 \( 1 + 4.34T + 29T^{2} \)
37 \( 1 + (-0.514 + 0.296i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.64 - 3.83i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.15 - 3.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.255iT - 47T^{2} \)
53 \( 1 + (-1.64 + 2.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.81 - 5.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 5.31T + 61T^{2} \)
67 \( 1 + (-8.18 - 4.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.83 + 3.94i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.249 + 0.143i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.03 + 3.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.3 + 7.11i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.37iT - 89T^{2} \)
97 \( 1 - 1.70iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.31974012755364835831015426340, −10.24131142787736519997256620513, −9.666083452564324927129186917193, −9.191642902653317371010036581502, −7.77390890232875416385042970963, −6.71593542259883079067263560817, −5.65922228582327746547573819022, −4.37181462400946472593029865903, −3.00877708670082540393526891248, −2.54498136471627485654693236746, 1.43944633838947416126115854152, 2.49897404006484079702461088832, 3.68192858755818673146240893826, 5.77898214769041998468232575047, 6.62913011243414913635555552117, 7.22092903318976515348472884992, 8.033292401626173563915583878268, 9.166377479336540347062950502283, 10.03945934335897680297435375155, 11.25685583521435033665081321307

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