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Properties

Label 2-351-39.32-c1-0-4
Degree $2$
Conductor $351$
Sign $-0.482 - 0.875i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (−2.64 + 0.708i)2-s + (4.74 − 2.74i)4-s + (2.27 + 2.27i)5-s + (−0.703 + 2.62i)7-s + (−6.73 + 6.73i)8-s + (−7.63 − 4.40i)10-s + (−0.281 − 1.05i)11-s + (1.86 + 3.08i)13-s − 7.43i·14-s + (7.54 − 13.0i)16-s + (0.897 + 1.55i)17-s + (−2.03 − 0.545i)19-s + (17.0 + 4.57i)20-s + (1.48 + 2.58i)22-s + (2.23 − 3.86i)23-s + ⋯
L(s)  = 1  + (−1.86 + 0.500i)2-s + (2.37 − 1.37i)4-s + (1.01 + 1.01i)5-s + (−0.265 + 0.991i)7-s + (−2.38 + 2.38i)8-s + (−2.41 − 1.39i)10-s + (−0.0850 − 0.317i)11-s + (0.516 + 0.856i)13-s − 1.98i·14-s + (1.88 − 3.26i)16-s + (0.217 + 0.376i)17-s + (−0.466 − 0.125i)19-s + (3.81 + 1.02i)20-s + (0.317 + 0.550i)22-s + (0.465 − 0.805i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.482 - 0.875i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.482 - 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.316911 + 0.536427i\)
\(L(\frac12)\) \(\approx\) \(0.316911 + 0.536427i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-1.86 - 3.08i)T \)
good2 \( 1 + (2.64 - 0.708i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-2.27 - 2.27i)T + 5iT^{2} \)
7 \( 1 + (0.703 - 2.62i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.281 + 1.05i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.897 - 1.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.03 + 0.545i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.23 + 3.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.22 + 1.86i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.95 - 4.95i)T - 31iT^{2} \)
37 \( 1 + (-4.41 + 1.18i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.53 - 0.410i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.73 - 3.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.476 - 0.476i)T - 47iT^{2} \)
53 \( 1 - 13.5iT - 53T^{2} \)
59 \( 1 + (-8.60 - 2.30i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.0624 + 0.108i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.58 + 9.65i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.23 + 4.60i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.79 - 3.79i)T + 73iT^{2} \)
79 \( 1 - 4.15T + 79T^{2} \)
83 \( 1 + (-6.36 - 6.36i)T + 83iT^{2} \)
89 \( 1 + (-2.50 - 9.33i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (16.4 + 4.40i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.13268601014914458469680551206, −10.70060177707327014413079696020, −9.690272362828694064680122017563, −9.092047443682110564508894389791, −8.299477161551007650946738242191, −7.00407416692328234554486711128, −6.32852015028623226313949921524, −5.68595766888221925882544859752, −2.80214730035270389145585932768, −1.82510747977656921455853197487, 0.78540551640757112791162929514, 1.95439487290216573469909029417, 3.56355456988957358899732880201, 5.51236568214429580389726289638, 6.78215306662529023834810945107, 7.71847834954624984417413560620, 8.611083017353279394447725295283, 9.490099837309999119477026980897, 10.01101466518142875245285638656, 10.79603269945602856226048462308

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