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Properties

Label 2-351-117.25-c1-0-4
Degree $2$
Conductor $351$
Sign $0.645 + 0.764i$
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  + (−0.784 − 0.452i)2-s + (−0.589 − 1.02i)4-s + (1.94 − 1.12i)5-s + (2.97 + 1.71i)7-s + 2.87i·8-s − 2.03·10-s + (3.20 + 1.84i)11-s + (−3.28 − 1.48i)13-s + (−1.55 − 2.69i)14-s + (0.124 − 0.214i)16-s + 4.21·17-s − 4.25i·19-s + (−2.29 − 1.32i)20-s + (−1.67 − 2.89i)22-s + (−1.89 − 3.27i)23-s + ⋯
L(s)  = 1  + (−0.554 − 0.320i)2-s + (−0.294 − 0.510i)4-s + (0.868 − 0.501i)5-s + (1.12 + 0.649i)7-s + 1.01i·8-s − 0.642·10-s + (0.965 + 0.557i)11-s + (−0.910 − 0.413i)13-s + (−0.415 − 0.720i)14-s + (0.0310 − 0.0537i)16-s + 1.02·17-s − 0.975i·19-s + (−0.512 − 0.295i)20-s + (−0.356 − 0.618i)22-s + (−0.394 − 0.683i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.645 + 0.764i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.645 + 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09838 - 0.510173i\)
\(L(\frac12)\) \(\approx\) \(1.09838 - 0.510173i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.28 + 1.48i)T \)
good2 \( 1 + (0.784 + 0.452i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.94 + 1.12i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.97 - 1.71i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.20 - 1.84i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.21T + 17T^{2} \)
19 \( 1 + 4.25iT - 19T^{2} \)
23 \( 1 + (1.89 + 3.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.18 + 2.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.37 + 3.67i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.49iT - 37T^{2} \)
41 \( 1 + (6.86 - 3.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.450 - 0.779i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.80 + 2.77i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.59T + 53T^{2} \)
59 \( 1 + (-4.44 + 2.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.50 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.7 - 6.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.65iT - 71T^{2} \)
73 \( 1 - 5.45iT - 73T^{2} \)
79 \( 1 + (5.46 - 9.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.465 - 0.268i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.75iT - 89T^{2} \)
97 \( 1 + (-5.87 - 3.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.43049622351009404956007414134, −10.00781740131006335125625737425, −9.733780533680294999153899618270, −8.709338063853031720106397250258, −7.971577312975973837930524310026, −6.39684194454605860875588438560, −5.23576353003560976080009547049, −4.72447439800728141038527067025, −2.37057521077812288165049224525, −1.31197293570130415918591721541, 1.52605917148988961438212668211, 3.39903831702727572608565593358, 4.58476765581932281964083351324, 5.96388879123457579508183422730, 7.06984856597308154671319677365, 7.84144991137973815890156833784, 8.753666846633160094234879490511, 9.811018987526389263273470926085, 10.35754210459859633131116651244, 11.66957394504873444881185032769

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