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Properties

Label 2-1274-7.4-c1-0-12
Degree $2$
Conductor $1274$
Sign $-0.991 + 0.126i$
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 + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.5 + 2.59i)5-s − 0.999·6-s − 0.999·8-s + (1 + 1.73i)9-s + (−1.5 + 2.59i)10-s + (−3 + 5.19i)11-s + (−0.499 − 0.866i)12-s + 13-s − 3·15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + (−1 + 1.73i)18-s + (−1 − 1.73i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s − 0.408·6-s − 0.353·8-s + (0.333 + 0.577i)9-s + (−0.474 + 0.821i)10-s + (−0.904 + 1.56i)11-s + (−0.144 − 0.249i)12-s + 0.277·13-s − 0.774·15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.235 + 0.408i)18-s + (−0.229 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (1145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.753826915\)
\(L(\frac12)\) \(\approx\) \(1.753826915\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.02414614354373510125618424346, −9.642144663484840962972623906659, −8.245821431173540695722316945748, −7.40211630618632686499206078777, −6.83349163197426809178972307291, −5.94716665172481786328718230579, −4.97070784315375346081156658939, −4.44416306589418593545210854697, −3.00555991530369871067846111712, −2.15382621102008754643720611810, 0.69911432621316140419650129518, 1.53712857088725695217922873413, 2.93100660601375220566987741143, 4.00602453723288286774536932848, 5.10217242807947181692504087355, 5.85522540598228304962787716990, 6.34418893371332079710871552929, 7.80803129834359095405016379425, 8.601675652314400376932627301865, 9.214793479021470869671545744802

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