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Properties

Label 2-1274-13.9-c1-0-4
Degree $2$
Conductor $1274$
Sign $-0.872 - 0.488i$
Analytic cond. $10.1729$
Root an. cond. $3.18950$
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  + (0.5 − 0.866i)2-s + (−1 + 1.73i)3-s + (−0.499 − 0.866i)4-s − 5-s + (0.999 + 1.73i)6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−1 + 1.73i)11-s + 1.99·12-s + (3.5 + 0.866i)13-s + (1 − 1.73i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 0.999·18-s + (−2 − 3.46i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.577 + 0.999i)3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (0.408 + 0.707i)6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.301 + 0.522i)11-s + 0.577·12-s + (0.970 + 0.240i)13-s + (0.258 − 0.447i)15-s + (−0.125 + 0.216i)16-s + (0.121 + 0.210i)17-s − 0.235·18-s + (−0.458 − 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.872 - 0.488i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4491209674\)
\(L(\frac12)\) \(\approx\) \(0.4491209674\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
13 \( 1 + (-3.5 - 0.866i)T \)
good3 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + T + 5T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8 + 13.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.12842709470517101469341650647, −9.536463339085512807806847576434, −8.580682746550486132331902009565, −7.64144302137958428729372525222, −6.44765613366715271895548277281, −5.59723642764747631880496381164, −4.66040623233785798307142382233, −4.13190480851434408048630445343, −3.18421643962415879015537718316, −1.73454836228826189828379742647, 0.17865777714988229945583735707, 1.68259153913701527687867897105, 3.31699995959704417592949731884, 4.14633558772711405987839317404, 5.53300937685864928819558553193, 5.94918098341926586812645733292, 6.85994492753932318680854354558, 7.53012921509658335110301234581, 8.281251386200839132678898002142, 8.991161347632169865956656976448

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