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Properties

Label 2-402-201.11-c1-0-20
Degree $2$
Conductor $402$
Sign $-0.868 - 0.495i$
Analytic cond. $3.20998$
Root an. cond. $1.79164$
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  + (−0.327 − 0.945i)2-s + (−0.685 − 1.59i)3-s + (−0.786 + 0.618i)4-s + (0.798 − 0.234i)5-s + (−1.27 + 1.16i)6-s + (0.0582 − 0.302i)7-s + (0.841 + 0.540i)8-s + (−2.06 + 2.18i)9-s + (−0.482 − 0.677i)10-s + (−4.16 − 3.96i)11-s + (1.52 + 0.826i)12-s + (−6.10 + 0.290i)13-s + (−0.304 + 0.0438i)14-s + (−0.920 − 1.10i)15-s + (0.235 − 0.971i)16-s + (2.05 − 2.61i)17-s + ⋯
L(s)  = 1  + (−0.231 − 0.668i)2-s + (−0.395 − 0.918i)3-s + (−0.393 + 0.309i)4-s + (0.357 − 0.104i)5-s + (−0.522 + 0.476i)6-s + (0.0220 − 0.114i)7-s + (0.297 + 0.191i)8-s + (−0.686 + 0.726i)9-s + (−0.152 − 0.214i)10-s + (−1.25 − 1.19i)11-s + (0.439 + 0.238i)12-s + (−1.69 + 0.0806i)13-s + (−0.0814 + 0.0117i)14-s + (−0.237 − 0.286i)15-s + (0.0589 − 0.242i)16-s + (0.499 − 0.634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(402\)    =    \(2 \cdot 3 \cdot 67\)
Sign: $-0.868 - 0.495i$
Analytic conductor: \(3.20998\)
Root analytic conductor: \(1.79164\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{402} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 402,\ (\ :1/2),\ -0.868 - 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132701 + 0.500026i\)
\(L(\frac12)\) \(\approx\) \(0.132701 + 0.500026i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.327 + 0.945i)T \)
3 \( 1 + (0.685 + 1.59i)T \)
67 \( 1 + (7.54 + 3.17i)T \)
good5 \( 1 + (-0.798 + 0.234i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-0.0582 + 0.302i)T + (-6.49 - 2.60i)T^{2} \)
11 \( 1 + (4.16 + 3.96i)T + (0.523 + 10.9i)T^{2} \)
13 \( 1 + (6.10 - 0.290i)T + (12.9 - 1.23i)T^{2} \)
17 \( 1 + (-2.05 + 2.61i)T + (-4.00 - 16.5i)T^{2} \)
19 \( 1 + (-3.85 + 0.742i)T + (17.6 - 7.06i)T^{2} \)
23 \( 1 + (0.0210 - 0.220i)T + (-22.5 - 4.35i)T^{2} \)
29 \( 1 + (9.13 - 5.27i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.90 + 0.0907i)T + (30.8 + 2.94i)T^{2} \)
37 \( 1 + (-4.87 + 8.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.34 - 1.33i)T + (29.6 - 28.2i)T^{2} \)
43 \( 1 + (-6.59 - 0.947i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (1.57 + 1.12i)T + (15.3 + 44.4i)T^{2} \)
53 \( 1 + (1.44 + 10.0i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-5.27 + 8.21i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (-2.31 - 2.42i)T + (-2.90 + 60.9i)T^{2} \)
71 \( 1 + (4.07 + 5.18i)T + (-16.7 + 68.9i)T^{2} \)
73 \( 1 + (1.08 - 1.03i)T + (3.47 - 72.9i)T^{2} \)
79 \( 1 + (-5.79 + 11.2i)T + (-45.8 - 64.3i)T^{2} \)
83 \( 1 + (-5.43 - 1.31i)T + (73.7 + 38.0i)T^{2} \)
89 \( 1 + (10.1 + 4.64i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 + (-6.05 - 3.49i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.95342873189884887466490736841, −9.909429482706570240701357449610, −9.037807809961095969294403487801, −7.68311645960123014974003466820, −7.38984176958601664191239432027, −5.65182942252397507988586904449, −5.15256358659299236421245610850, −3.16214455803005690427849207540, −2.10059560681045786977386659770, −0.35808715210197250303947573766, 2.49511515200505340438322579600, 4.23064721113903659018377373020, 5.21623339348509566133229011215, 5.82371766064045175665135089999, 7.27324389555478866756822489246, 7.927953406931557961765409114806, 9.400472883099915529028349228365, 9.906254751169429310080914694713, 10.44263907137047180622871325959, 11.78039728371121713493082930663

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