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Properties

Label 2-416-104.19-c1-0-7
Degree $2$
Conductor $416$
Sign $0.972 + 0.231i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
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  + (0.223 − 0.386i)3-s + (−0.612 − 0.612i)5-s + (2.00 + 0.537i)7-s + (1.40 + 2.42i)9-s + (3.04 − 0.816i)11-s + (−3.45 − 1.02i)13-s + (−0.373 + 0.100i)15-s + (2.68 − 1.55i)17-s + (3.53 + 0.948i)19-s + (0.655 − 0.655i)21-s + (1.56 − 2.70i)23-s − 4.24i·25-s + 2.58·27-s + (6.13 + 3.54i)29-s + (−2.77 − 2.77i)31-s + ⋯
L(s)  = 1  + (0.128 − 0.223i)3-s + (−0.274 − 0.274i)5-s + (0.758 + 0.203i)7-s + (0.466 + 0.808i)9-s + (0.918 − 0.246i)11-s + (−0.958 − 0.283i)13-s + (−0.0964 + 0.0258i)15-s + (0.651 − 0.376i)17-s + (0.812 + 0.217i)19-s + (0.143 − 0.143i)21-s + (0.325 − 0.563i)23-s − 0.849i·25-s + 0.497·27-s + (1.13 + 0.657i)29-s + (−0.498 − 0.498i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.972 + 0.231i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.972 + 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56515 - 0.183987i\)
\(L(\frac12)\) \(\approx\) \(1.56515 - 0.183987i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (3.45 + 1.02i)T \)
good3 \( 1 + (-0.223 + 0.386i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.612 + 0.612i)T + 5iT^{2} \)
7 \( 1 + (-2.00 - 0.537i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.04 + 0.816i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.68 + 1.55i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.53 - 0.948i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.56 + 2.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.13 - 3.54i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.77 + 2.77i)T + 31iT^{2} \)
37 \( 1 + (0.155 + 0.580i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.60 - 5.98i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.34 - 3.08i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.19 - 6.19i)T - 47iT^{2} \)
53 \( 1 - 2.19iT - 53T^{2} \)
59 \( 1 + (1.67 - 6.25i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.346 + 0.199i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.40 + 8.98i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.33 + 16.1i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.53 + 6.53i)T + 73iT^{2} \)
79 \( 1 - 12.9iT - 79T^{2} \)
83 \( 1 + (9.22 - 9.22i)T - 83iT^{2} \)
89 \( 1 + (6.08 - 1.62i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (15.1 + 4.07i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.29871843418456045845888548079, −10.26004541864723758229179770849, −9.364110040422578806627685786653, −8.212416966297327009278373164039, −7.68508560775830851141880177520, −6.58519463675182895629420143452, −5.17426342275804927652395076663, −4.48803804198555070520764574759, −2.88470963708672234920547046008, −1.37344641690306194952559351537, 1.45007319665933950138927253952, 3.28192474755651532634963529183, 4.27179277743187263515410008059, 5.34388766243379653224521765932, 6.81080893520165300282656097253, 7.38148179989101068316719898747, 8.564339647665530194706464304380, 9.557248352843997550173033090843, 10.17049612628388713418499636237, 11.47918290541830145841476747090

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