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Properties

Label 2-1274-13.3-c1-0-18
Degree $2$
Conductor $1274$
Sign $0.477 - 0.878i$
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 + 3·5-s + (0.999 − 1.73i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)10-s + (3 + 5.19i)11-s + 1.99·12-s + (−2.5 + 2.59i)13-s + (−3 − 5.19i)15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)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 + 1.34·5-s + (0.408 − 0.707i)6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + (0.904 + 1.56i)11-s + 0.577·12-s + (−0.693 + 0.720i)13-s + (−0.774 − 1.34i)15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s − 0.235·18-s + (−0.458 + 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.943830207\)
\(L(\frac12)\) \(\approx\) \(1.943830207\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
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 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 18T + 83T^{2} \)
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

−9.592790317431992370880503539290, −9.183043598323599791870562301793, −7.83659629109098379002609278432, −7.04305710099504929732656062375, −6.48176814247740568904419244753, −5.94838412085002173454175268380, −4.93707233038645688216676052433, −4.01756702533664978641978711766, −2.19643727923927030952285789229, −1.53939441476362889393714968644, 0.815486867648434296659770599691, 2.42460239638033558749577573095, 3.29262207942655981827696509560, 4.64546906601358903047720777159, 5.05059501891673975283908694029, 6.08862392658530310888661844535, 6.51295273266924590585757020607, 8.193069793412109045316997730934, 9.315124693233739628636875631567, 9.497863034028464987564868850804

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