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Properties

Label 2-1274-91.9-c1-0-37
Degree $2$
Conductor $1274$
Sign $0.263 + 0.964i$
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 − 3-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s + (−0.5 − 0.866i)6-s − 0.999·8-s − 2·9-s + 3·10-s + (0.499 − 0.866i)12-s + (−2.5 + 2.59i)13-s + (−1.5 + 2.59i)15-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (−1 − 1.73i)18-s + 4·19-s + (1.50 + 2.59i)20-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s + (−0.204 − 0.353i)6-s − 0.353·8-s − 0.666·9-s + 0.948·10-s + (0.144 − 0.249i)12-s + (−0.693 + 0.720i)13-s + (−0.387 + 0.670i)15-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (−0.235 − 0.408i)18-s + 0.917·19-s + (0.335 + 0.580i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.263 + 0.964i$
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: \((2,\ 1274,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.120645306\)
\(L(\frac12)\) \(\approx\) \(1.120645306\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 2.59i)T \)
good3 \( 1 + T + 3T^{2} \)
5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 11T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
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

−9.254034807673630436894368580215, −8.887582598650418436571681474823, −7.72205730629080281623230381787, −6.99019899994442451671999264509, −5.89872117450021680048352982695, −5.25702039696606771720523142484, −4.87835043638442368980489008430, −3.53103876129170454823743345900, −2.10034957828567162865403500753, −0.44118633746432247382182428817, 1.54486439915861621060276418481, 2.86300214056573156606818326103, 3.38819951979282175636496669444, 4.88477375169730820180653549122, 5.76233916570331437820209514018, 6.16796308775328102581971633125, 7.27402234304988270281171661731, 8.182848606555343385429160277678, 9.405432522183052479825092988294, 10.12530578761850590785083600903

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