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Properties

Label 2-399-133.40-c1-0-24
Degree $2$
Conductor $399$
Sign $-0.971 + 0.237i$
Analytic cond. $3.18603$
Root an. cond. $1.78494$
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  + (1.77 − 2.11i)2-s + (−0.173 + 0.984i)3-s + (−0.981 − 5.56i)4-s + (−1.01 − 0.179i)5-s + (1.77 + 2.11i)6-s + (−2.26 − 1.36i)7-s + (−8.75 − 5.05i)8-s + (−0.939 − 0.342i)9-s + (−2.18 + 1.83i)10-s − 0.467·11-s + 5.65·12-s + (4.39 − 3.69i)13-s + (−6.91 + 2.38i)14-s + (0.352 − 0.968i)15-s + (−15.6 + 5.69i)16-s + (1.60 + 4.39i)17-s + ⋯
L(s)  = 1  + (1.25 − 1.49i)2-s + (−0.100 + 0.568i)3-s + (−0.490 − 2.78i)4-s + (−0.454 − 0.0800i)5-s + (0.725 + 0.865i)6-s + (−0.857 − 0.514i)7-s + (−3.09 − 1.78i)8-s + (−0.313 − 0.114i)9-s + (−0.690 + 0.579i)10-s − 0.140·11-s + 1.63·12-s + (1.21 − 1.02i)13-s + (−1.84 + 0.637i)14-s + (0.0910 − 0.250i)15-s + (−3.91 + 1.42i)16-s + (0.388 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $-0.971 + 0.237i$
Analytic conductor: \(3.18603\)
Root analytic conductor: \(1.78494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :1/2),\ -0.971 + 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.230817 - 1.91402i\)
\(L(\frac12)\) \(\approx\) \(0.230817 - 1.91402i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (2.26 + 1.36i)T \)
19 \( 1 + (-4.00 + 1.72i)T \)
good2 \( 1 + (-1.77 + 2.11i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.01 + 0.179i)T + (4.69 + 1.71i)T^{2} \)
11 \( 1 + 0.467T + 11T^{2} \)
13 \( 1 + (-4.39 + 3.69i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.60 - 4.39i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-4.77 + 4.01i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-6.63 + 1.16i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.11 - 3.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.40 - 1.38i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.03 - 1.70i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (7.48 - 2.72i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.0867 + 0.238i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-4.31 + 0.760i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-3.23 + 1.17i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.71 + 4.43i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.83 + 3.37i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.30 - 6.33i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (-5.64 - 0.995i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (5.15 + 14.1i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (4.82 - 2.78i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.86 - 10.5i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-0.841 + 4.77i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.83670744811851054311522559387, −10.39475193191604955978488198780, −9.602100130010170740402405266138, −8.407829310218354593252535995879, −6.52842645471434727616984774998, −5.65697202229672639035419967097, −4.57594825930765826920887532812, −3.59017734611098692369746335789, −3.02066648481999370369221254917, −0.915190648700101414708476967909, 2.98577464429223504480911298127, 3.86517714626262619829822622308, 5.24408574254427702868535423344, 6.01718106948857247035564000353, 6.90154903950363279474672402386, 7.52168520273668437918512007569, 8.576072668954378685589552745941, 9.399499055239371007990449856355, 11.55401918159884563394233285363, 11.85371217229725071862071402088

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