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Properties

Label 2-352-44.39-c1-0-8
Degree $2$
Conductor $352$
Sign $0.374 + 0.927i$
Analytic cond. $2.81073$
Root an. cond. $1.67652$
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  + (1.68 − 2.31i)3-s + (−0.612 + 1.88i)5-s + (1.05 − 0.767i)7-s + (−1.60 − 4.92i)9-s + (0.361 − 3.29i)11-s + (3.77 − 1.22i)13-s + (3.33 + 4.58i)15-s + (−5.89 − 1.91i)17-s + (4.95 + 3.60i)19-s − 3.73i·21-s + 0.399i·23-s + (0.865 + 0.628i)25-s + (−5.92 − 1.92i)27-s + (−0.0810 − 0.111i)29-s + (−8.27 + 2.69i)31-s + ⋯
L(s)  = 1  + (0.970 − 1.33i)3-s + (−0.273 + 0.843i)5-s + (0.399 − 0.290i)7-s + (−0.533 − 1.64i)9-s + (0.109 − 0.994i)11-s + (1.04 − 0.340i)13-s + (0.860 + 1.18i)15-s + (−1.42 − 0.464i)17-s + (1.13 + 0.826i)19-s − 0.815i·21-s + 0.0833i·23-s + (0.173 + 0.125i)25-s + (−1.14 − 0.370i)27-s + (−0.0150 − 0.0207i)29-s + (−1.48 + 0.483i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48641 - 1.00263i\)
\(L(\frac12)\) \(\approx\) \(1.48641 - 1.00263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.361 + 3.29i)T \)
good3 \( 1 + (-1.68 + 2.31i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.612 - 1.88i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.05 + 0.767i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-3.77 + 1.22i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (5.89 + 1.91i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.95 - 3.60i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.399iT - 23T^{2} \)
29 \( 1 + (0.0810 + 0.111i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (8.27 - 2.69i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.18 - 5.94i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.410 - 0.565i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.83T + 43T^{2} \)
47 \( 1 + (-4.56 + 6.28i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.99 - 12.2i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.53 - 2.11i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-6.26 - 2.03i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 1.32iT - 67T^{2} \)
71 \( 1 + (9.26 + 3.00i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.92 - 2.65i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.49 - 4.60i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.12 + 6.54i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + (-1.61 - 4.96i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.30127668424775360993376870963, −10.65207892978607389646764280802, −9.032124675174325219469993726559, −8.416862168767850870047210888322, −7.43584441387303098825310482110, −6.85937571462708223667626968415, −5.73734917045971802847963871680, −3.70450492321192552348867049169, −2.85136042299985997275459948124, −1.35218177361844790472486595473, 2.11132383619404325593358980191, 3.72726708597064259979784033185, 4.47645281235036207352323172469, 5.33990265887801451872275509709, 7.08450721626288237400835496471, 8.378883014054627316405320231999, 8.946412648442292148917166155194, 9.487713418061466397869869312751, 10.68706173145832306109225977846, 11.41380218330147265540350693302

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