Properties

Label 2-1260-140.27-c0-0-3
Degree $2$
Conductor $1260$
Sign $-0.229 + 0.973i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + 5-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s − 1.41i·11-s − 1.00·14-s − 1.00·16-s + (−1 + i)17-s + 1.41i·19-s − 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (−0.707 + 0.707i)28-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + 5-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s − 1.41i·11-s − 1.00·14-s − 1.00·16-s + (−1 + i)17-s + 1.41i·19-s − 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (−0.707 + 0.707i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.594554252\)
\(L(\frac12)\) \(\approx\) \(1.594554252\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + 2iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.985060706171560795946849417372, −9.059483063031849444247540840576, −8.226702133397213320268038174334, −6.70837816087971111233251865946, −6.16611103227721296195916685166, −5.55880597242699760921462607712, −4.28167444796363657382045257402, −3.49025780032061954018839079916, −2.48777114432843828978640099774, −1.19643705564035640918987382617, 2.33572953309973904024194901449, 2.85456487422485691404274893742, 4.53352003046742537680602568600, 4.96310940970327705418246472101, 6.09118537405788621281013089458, 6.66352544892860029015835683131, 7.32089807297289391367550641204, 8.538479576660980878176457266066, 9.420743069413805176751368100991, 9.668279572763449269209909433046

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