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Properties

Label 2-351-117.94-c1-0-0
Degree $2$
Conductor $351$
Sign $-0.994 - 0.101i$
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  + (−0.433 + 0.751i)2-s + (0.623 + 1.08i)4-s + (0.0324 − 0.0561i)5-s − 3.92·7-s − 2.81·8-s + (0.0281 + 0.0486i)10-s + (−2.64 + 4.58i)11-s + (−0.188 − 3.60i)13-s + (1.70 − 2.94i)14-s + (−0.0259 + 0.0449i)16-s + (−2.28 + 3.95i)17-s + (0.281 − 0.486i)19-s + 0.0808·20-s + (−2.29 − 3.97i)22-s + 2.85·23-s + ⋯
L(s)  = 1  + (−0.306 + 0.531i)2-s + (0.311 + 0.540i)4-s + (0.0144 − 0.0251i)5-s − 1.48·7-s − 0.995·8-s + (0.00888 + 0.0153i)10-s + (−0.797 + 1.38i)11-s + (−0.0523 − 0.998i)13-s + (0.454 − 0.787i)14-s + (−0.00649 + 0.0112i)16-s + (−0.553 + 0.958i)17-s + (0.0644 − 0.111i)19-s + 0.0180·20-s + (−0.489 − 0.847i)22-s + 0.595·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.994 - 0.101i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.994 - 0.101i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0299972 + 0.588586i\)
\(L(\frac12)\) \(\approx\) \(0.0299972 + 0.588586i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.188 + 3.60i)T \)
good2 \( 1 + (0.433 - 0.751i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.0324 + 0.0561i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.92T + 7T^{2} \)
11 \( 1 + (2.64 - 4.58i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.28 - 3.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.281 + 0.486i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + (-3.00 + 5.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.23 - 7.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.506 + 0.877i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.34T + 41T^{2} \)
43 \( 1 + 6.90T + 43T^{2} \)
47 \( 1 + (-2.22 - 3.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.68T + 53T^{2} \)
59 \( 1 + (-4.57 - 7.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 6.35T + 61T^{2} \)
67 \( 1 - 3.53T + 67T^{2} \)
71 \( 1 + (-5.02 + 8.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.39T + 73T^{2} \)
79 \( 1 + (-5.67 - 9.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.87 - 3.24i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.00 - 3.47i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.35T + 97T^{2} \)
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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

−12.25052574986430279008380690882, −10.81155869422842516302559128047, −9.987804498739261875939412378207, −9.097189846738600546511724827696, −8.066458239375401157335842794599, −7.11593943620563314671103767987, −6.48254087886288678173150495231, −5.24249679964231507000380766086, −3.60624036158311131416985183492, −2.58477951280901482887254819691, 0.41166634420239290187953162675, 2.48915902886489361284377165728, 3.41345807678021327324162736362, 5.21489937090227223839068197181, 6.31797024908348521895476135871, 6.93012694731025643817072532856, 8.596876821860428819505863469502, 9.357903145955369634475170570000, 10.12641648641252334290163063781, 11.01120219810949733279333540197

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