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Properties

Label 2-351-39.5-c1-0-1
Degree $2$
Conductor $351$
Sign $0.893 - 0.449i$
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  + (−1.04 − 1.04i)2-s + 0.180i·4-s + (1.27 + 1.27i)5-s + (2.33 + 2.33i)7-s + (−1.90 + 1.90i)8-s − 2.65i·10-s + (−3.47 + 3.47i)11-s + (0.619 + 3.55i)13-s − 4.87i·14-s + 4.32·16-s − 3.61·17-s + (−4.11 + 4.11i)19-s + (−0.229 + 0.229i)20-s + 7.26·22-s + 7.41·23-s + ⋯
L(s)  = 1  + (−0.738 − 0.738i)2-s + 0.0900i·4-s + (0.569 + 0.569i)5-s + (0.881 + 0.881i)7-s + (−0.671 + 0.671i)8-s − 0.840i·10-s + (−1.04 + 1.04i)11-s + (0.171 + 0.985i)13-s − 1.30i·14-s + 1.08·16-s − 0.877·17-s + (−0.943 + 0.943i)19-s + (−0.0512 + 0.0512i)20-s + 1.54·22-s + 1.54·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.893 - 0.449i$
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: \((2,\ 351,\ (\ :1/2),\ 0.893 - 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.883211 + 0.209909i\)
\(L(\frac12)\) \(\approx\) \(0.883211 + 0.209909i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.619 - 3.55i)T \)
good2 \( 1 + (1.04 + 1.04i)T + 2iT^{2} \)
5 \( 1 + (-1.27 - 1.27i)T + 5iT^{2} \)
7 \( 1 + (-2.33 - 2.33i)T + 7iT^{2} \)
11 \( 1 + (3.47 - 3.47i)T - 11iT^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
19 \( 1 + (4.11 - 4.11i)T - 19iT^{2} \)
23 \( 1 - 7.41T + 23T^{2} \)
29 \( 1 + 0.458iT - 29T^{2} \)
31 \( 1 + (-4.25 + 4.25i)T - 31iT^{2} \)
37 \( 1 + (3.35 + 3.35i)T + 37iT^{2} \)
41 \( 1 + (-3.47 - 3.47i)T + 41iT^{2} \)
43 \( 1 - 4.90iT - 43T^{2} \)
47 \( 1 + (-8.36 + 8.36i)T - 47iT^{2} \)
53 \( 1 + 8.48iT - 53T^{2} \)
59 \( 1 + (-0.117 + 0.117i)T - 59iT^{2} \)
61 \( 1 + 6.49T + 61T^{2} \)
67 \( 1 + (2.08 - 2.08i)T - 67iT^{2} \)
71 \( 1 + (-8.69 - 8.69i)T + 71iT^{2} \)
73 \( 1 + (-2.50 - 2.50i)T + 73iT^{2} \)
79 \( 1 - 4.07T + 79T^{2} \)
83 \( 1 + (8.79 + 8.79i)T + 83iT^{2} \)
89 \( 1 + (2.66 - 2.66i)T - 89iT^{2} \)
97 \( 1 + (2.72 - 2.72i)T - 97iT^{2} \)
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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

−11.28698646848013015567766139773, −10.62213012723787104218568244218, −9.830066973699350564452897506188, −8.934457209470489346975002659843, −8.158425807553927293200447046604, −6.78200524777919289263869095164, −5.68133568387324432044115726388, −4.60549966403993002135314025064, −2.46549710179166364932150278314, −1.96152753393635012458125881323, 0.809017025028034934033300748625, 2.97285541851702604336647345513, 4.64344677881896662644630836444, 5.65555989198650323903481767558, 6.86228434257218741688866253892, 7.78182693664430114379559574774, 8.572954932713810153537419984743, 9.152217006497815902540236046107, 10.68822663381103517045311523172, 10.86876557729165270295952780925

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