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Properties

Label 2-1274-13.4-c1-0-25
Degree $2$
Conductor $1274$
Sign $0.711 - 0.702i$
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  + (−0.866 − 0.5i)2-s + (−1.36 + 2.36i)3-s + (0.499 + 0.866i)4-s + i·5-s + (2.36 − 1.36i)6-s − 0.999i·8-s + (−2.23 − 3.86i)9-s + (0.5 − 0.866i)10-s + (2.36 + 1.36i)11-s − 2.73·12-s + (2.59 − 2.5i)13-s + (−2.36 − 1.36i)15-s + (−0.5 + 0.866i)16-s + (1.13 + 1.96i)17-s + 4.46i·18-s + (4.73 − 2.73i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.788 + 1.36i)3-s + (0.249 + 0.433i)4-s + 0.447i·5-s + (0.965 − 0.557i)6-s − 0.353i·8-s + (−0.744 − 1.28i)9-s + (0.158 − 0.273i)10-s + (0.713 + 0.411i)11-s − 0.788·12-s + (0.720 − 0.693i)13-s + (−0.610 − 0.352i)15-s + (−0.125 + 0.216i)16-s + (0.275 + 0.476i)17-s + 1.05i·18-s + (1.08 − 0.626i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.032374708\)
\(L(\frac12)\) \(\approx\) \(1.032374708\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 \)
13 \( 1 + (-2.59 + 2.5i)T \)
good3 \( 1 + (1.36 - 2.36i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - iT - 5T^{2} \)
11 \( 1 + (-2.36 - 1.36i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.13 - 1.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.73 + 2.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.36 + 4.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.73iT - 31T^{2} \)
37 \( 1 + (-4.5 - 2.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.86 - 2.23i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.901 + 1.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 - 8.46T + 53T^{2} \)
59 \( 1 + (6.29 - 3.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.13 + 7.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.90 - 2.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (12 - 6.92i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.39iT - 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + (-8.19 - 4.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.26 + 0.732i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.00595332697119388421019031269, −9.221471089552621420989961107715, −8.450794117844588772084221945466, −7.33429593984877850925516367082, −6.37848850639581572429491967690, −5.56835945456943844201263443765, −4.50508194200424140892110062596, −3.72492378373812466345746494227, −2.72975081746811450121505036781, −0.854987506322031062500950403843, 1.01952172076970080916165886682, 1.49288526919103277691109533016, 3.20633551373305424985345138543, 4.79550186875561670873487602936, 5.76984925093302297641688383600, 6.32493819762885843300118873352, 7.21095419392996057731337118551, 7.67270844259261979308490062217, 8.822321688234281131677519305515, 9.212138167291152303792389366264

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