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Properties

Label 2-1274-91.81-c1-0-26
Degree $2$
Conductor $1274$
Sign $0.703 + 0.710i$
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 + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·8-s − 3·9-s + 0.999·10-s + 4·11-s + (−3.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + (1.5 − 2.59i)18-s + (−0.499 + 0.866i)20-s + (−2 + 3.46i)22-s + (2 − 3.46i)23-s + (2 − 3.46i)25-s + (1 − 3.46i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.353·8-s − 9-s + 0.316·10-s + 1.20·11-s + (−0.970 + 0.240i)13-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (0.353 − 0.612i)18-s + (−0.111 + 0.193i)20-s + (−0.426 + 0.738i)22-s + (0.417 − 0.722i)23-s + (0.400 − 0.692i)25-s + (0.196 − 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ 0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9486662364\)
\(L(\frac12)\) \(\approx\) \(0.9486662364\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (-4 + 6.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
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

−9.351964053554124795378698358521, −8.656188529841566323740662880622, −8.147632836067489378688335317932, −7.02572093811939281490106874204, −6.40724720025065079162952937128, −5.43322216959255624793171134783, −4.59629068514213009223124216308, −3.54023507096519843056312207290, −2.09923813210027357991416000185, −0.50605443589789670863271595371, 1.20503749148774732891790408622, 2.72606983128633489630764405152, 3.32862262084574237466263846843, 4.53743838747959346643601269595, 5.51120108154346794493574770742, 6.62639983009215043486445830464, 7.43030305762522662513686648369, 8.242106424307843831340100725985, 9.242088651756470189714322422692, 9.584072725956746237973349572380

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