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Properties

Label 2-351-39.5-c1-0-16
Degree $2$
Conductor $351$
Sign $0.376 + 0.926i$
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  + (1.14 + 1.14i)2-s + 0.599i·4-s + (−2.84 − 2.84i)5-s + (−2.82 − 2.82i)7-s + (1.59 − 1.59i)8-s − 6.48i·10-s + (−2.08 + 2.08i)11-s + (3.59 − 0.331i)13-s − 6.44i·14-s + 4.83·16-s − 3.94·17-s + (2.32 − 2.32i)19-s + (1.70 − 1.70i)20-s − 4.74·22-s + 0.755·23-s + ⋯
L(s)  = 1  + (0.806 + 0.806i)2-s + 0.299i·4-s + (−1.27 − 1.27i)5-s + (−1.06 − 1.06i)7-s + (0.564 − 0.564i)8-s − 2.05i·10-s + (−0.627 + 0.627i)11-s + (0.995 − 0.0918i)13-s − 1.72i·14-s + 1.20·16-s − 0.957·17-s + (0.532 − 0.532i)19-s + (0.381 − 0.381i)20-s − 1.01·22-s + 0.157·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.376 + 0.926i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06590 - 0.717369i\)
\(L(\frac12)\) \(\approx\) \(1.06590 - 0.717369i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 + 0.331i)T \)
good2 \( 1 + (-1.14 - 1.14i)T + 2iT^{2} \)
5 \( 1 + (2.84 + 2.84i)T + 5iT^{2} \)
7 \( 1 + (2.82 + 2.82i)T + 7iT^{2} \)
11 \( 1 + (2.08 - 2.08i)T - 11iT^{2} \)
17 \( 1 + 3.94T + 17T^{2} \)
19 \( 1 + (-2.32 + 2.32i)T - 19iT^{2} \)
23 \( 1 - 0.755T + 23T^{2} \)
29 \( 1 - 3.40iT - 29T^{2} \)
31 \( 1 + (-6.39 + 6.39i)T - 31iT^{2} \)
37 \( 1 + (2.85 + 2.85i)T + 37iT^{2} \)
41 \( 1 + (-2.08 - 2.08i)T + 41iT^{2} \)
43 \( 1 - 0.526iT - 43T^{2} \)
47 \( 1 + (-3.18 + 3.18i)T - 47iT^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + (-7.20 + 7.20i)T - 59iT^{2} \)
61 \( 1 - 2.91T + 61T^{2} \)
67 \( 1 + (5.13 - 5.13i)T - 67iT^{2} \)
71 \( 1 + (2.08 + 2.08i)T + 71iT^{2} \)
73 \( 1 + (3.56 + 3.56i)T + 73iT^{2} \)
79 \( 1 - 5.79T + 79T^{2} \)
83 \( 1 + (-7.18 - 7.18i)T + 83iT^{2} \)
89 \( 1 + (1.51 - 1.51i)T - 89iT^{2} \)
97 \( 1 + (-2.07 + 2.07i)T - 97iT^{2} \)
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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

−11.46187179576160653483430392605, −10.41917807024574307860421253505, −9.357634347593882319532233535963, −8.205995144072111714305581044832, −7.32756037535872843254113259916, −6.56507809989838323481170729374, −5.20643869568023611739936762536, −4.33759890563842219468653148386, −3.64015667909104544400858501524, −0.70109349887895909424328704339, 2.71295311584109760342583148728, 3.21146414264671442637358558256, 4.19619117376579742555071273601, 5.76671652294862497211786171100, 6.73076526489694647877063757753, 7.928268832357820668732243966106, 8.779911518939129874491731965749, 10.34250346959944240579849999879, 10.96090952545874967922080009365, 11.78013500404801624173885878423

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