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Properties

Label 2-351-117.43-c1-0-8
Degree $2$
Conductor $351$
Sign $0.905 + 0.423i$
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  + (−2.29 + 1.32i)2-s + (2.51 − 4.35i)4-s + (2.43 − 1.40i)5-s − 0.261i·7-s + 8.03i·8-s + (−3.72 + 6.44i)10-s + (1.34 − 0.779i)11-s + (−1.29 − 3.36i)13-s + (0.346 + 0.599i)14-s + (−5.62 − 9.73i)16-s + (−3.65 − 6.32i)17-s + (0.447 − 0.258i)19-s − 14.1i·20-s + (−2.06 + 3.57i)22-s − 2.74·23-s + ⋯
L(s)  = 1  + (−1.62 + 0.937i)2-s + (1.25 − 2.17i)4-s + (1.08 − 0.627i)5-s − 0.0987i·7-s + 2.84i·8-s + (−1.17 + 2.03i)10-s + (0.407 − 0.234i)11-s + (−0.360 − 0.932i)13-s + (0.0925 + 0.160i)14-s + (−1.40 − 2.43i)16-s + (−0.886 − 1.53i)17-s + (0.102 − 0.0592i)19-s − 3.15i·20-s + (−0.440 + 0.763i)22-s − 0.573·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.905 + 0.423i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.905 + 0.423i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671310 - 0.149185i\)
\(L(\frac12)\) \(\approx\) \(0.671310 - 0.149185i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 + 3.36i)T \)
good2 \( 1 + (2.29 - 1.32i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.43 + 1.40i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.261iT - 7T^{2} \)
11 \( 1 + (-1.34 + 0.779i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.65 + 6.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.447 + 0.258i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.74T + 23T^{2} \)
29 \( 1 + (-1.78 - 3.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.17 + 1.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.90 - 1.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.42iT - 41T^{2} \)
43 \( 1 + 3.49T + 43T^{2} \)
47 \( 1 + (-7.10 - 4.10i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.68T + 53T^{2} \)
59 \( 1 + (-9.64 - 5.57i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.61T + 61T^{2} \)
67 \( 1 + 8.21iT - 67T^{2} \)
71 \( 1 + (-5.40 + 3.12i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.76iT - 73T^{2} \)
79 \( 1 + (1.36 - 2.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.18 - 4.14i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.18 - 0.684i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−10.93856485775598376349894785314, −10.08640156546791812983454268253, −9.359038271113581579566091978995, −8.808592215840783097723707869896, −7.75178609007840514681879306006, −6.81313812841268921651524646571, −5.85874260348901912440439602387, −5.01857561333626154693398568898, −2.33516903801512213203621915982, −0.819685238618308896283796891591, 1.71864821213351559358808464616, 2.50402278125003669140451859802, 4.05106068358691490747354623932, 6.22173509087897672313789031551, 6.91857904702431989752626041397, 8.178502487160364214002229670283, 9.014386805205184020940083722062, 9.862216320111367915109483008467, 10.34047949015779843152612790954, 11.27268967551652966550226780307

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