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Properties

Label 2-1274-91.9-c1-0-2
Degree $2$
Conductor $1274$
Sign $-0.542 - 0.840i$
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 + 1.73·3-s + (−0.499 + 0.866i)4-s + (0.133 − 0.232i)5-s + (−0.866 − 1.49i)6-s + 0.999·8-s − 0.267·10-s − 5.46·11-s + (−0.866 + 1.49i)12-s + (−1.59 + 3.23i)13-s + (0.232 − 0.401i)15-s + (−0.5 − 0.866i)16-s + (−1.73 + 3i)17-s − 3.46·19-s + (0.133 + 0.232i)20-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + 1.00·3-s + (−0.249 + 0.433i)4-s + (0.0599 − 0.103i)5-s + (−0.353 − 0.612i)6-s + 0.353·8-s − 0.0847·10-s − 1.64·11-s + (−0.249 + 0.433i)12-s + (−0.443 + 0.896i)13-s + (0.0599 − 0.103i)15-s + (−0.125 − 0.216i)16-s + (−0.420 + 0.727i)17-s − 0.794·19-s + (0.0299 + 0.0518i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.542 - 0.840i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.542 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3371900908\)
\(L(\frac12)\) \(\approx\) \(0.3371900908\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 3.23i)T \)
good3 \( 1 - 1.73T + 3T^{2} \)
5 \( 1 + (-0.133 + 0.232i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
17 \( 1 + (1.73 - 3i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + (4.23 + 7.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.46 - 7.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.267 - 0.464i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 - 2.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.73 + 8.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.46 + 4.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.46 - 6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.40 + 2.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 3.19T + 61T^{2} \)
67 \( 1 - 4.92T + 67T^{2} \)
71 \( 1 + (-1.23 - 2.13i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.267 + 0.464i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.535 - 0.928i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (4.26 + 7.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.53 + 2.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.992971718910752325996887459109, −8.794434937748294970020875700145, −8.734066083251560119544792404775, −7.76460095393931607825230565886, −6.97446697537354976425870375436, −5.67090277695161467450078379881, −4.60671508026397272078508030620, −3.65031587250094535240699344116, −2.54349142232527665159336400466, −2.00851199434171652509463892538, 0.12346947926402638492537518159, 2.25498450144966026368959622866, 2.91272863432513014256786135942, 4.24683384115946361214579374017, 5.34944984296480373868747589633, 6.02082334231184081084864498769, 7.30162501584717986853354643735, 7.994542660540507147840149757711, 8.223400934569214529421385310176, 9.425506635760604835667970508964

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