Properties

Label 2-1260-28.27-c1-0-57
Degree $2$
Conductor $1260$
Sign $-0.572 + 0.819i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (−1.07 − 0.921i)2-s + (0.303 + 1.97i)4-s + i·5-s + (1.82 − 1.91i)7-s + (1.49 − 2.40i)8-s + (0.921 − 1.07i)10-s − 6.24i·11-s − 2.40i·13-s + (−3.72 + 0.373i)14-s + (−3.81 + 1.19i)16-s + 1.30i·17-s + 3.94·19-s + (−1.97 + 0.303i)20-s + (−5.74 + 6.69i)22-s + 3.55i·23-s + ⋯
L(s)  = 1  + (−0.758 − 0.651i)2-s + (0.151 + 0.988i)4-s + 0.447i·5-s + (0.690 − 0.723i)7-s + (0.528 − 0.848i)8-s + (0.291 − 0.339i)10-s − 1.88i·11-s − 0.666i·13-s + (−0.995 + 0.0997i)14-s + (−0.954 + 0.299i)16-s + 0.317i·17-s + 0.905·19-s + (−0.442 + 0.0677i)20-s + (−1.22 + 1.42i)22-s + 0.741i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.572 + 0.819i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.572 + 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9869863869\)
\(L(\frac12)\) \(\approx\) \(0.9869863869\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 + 0.921i)T \)
3 \( 1 \)
5 \( 1 - iT \)
7 \( 1 + (-1.82 + 1.91i)T \)
good11 \( 1 + 6.24iT - 11T^{2} \)
13 \( 1 + 2.40iT - 13T^{2} \)
17 \( 1 - 1.30iT - 17T^{2} \)
19 \( 1 - 3.94T + 19T^{2} \)
23 \( 1 - 3.55iT - 23T^{2} \)
29 \( 1 + 1.44T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 + 5.06T + 37T^{2} \)
41 \( 1 + 1.26iT - 41T^{2} \)
43 \( 1 + 2.19iT - 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 - 0.415T + 59T^{2} \)
61 \( 1 + 12.6iT - 61T^{2} \)
67 \( 1 + 3.10iT - 67T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + 1.64iT - 73T^{2} \)
79 \( 1 + 9.18iT - 79T^{2} \)
83 \( 1 - 7.39T + 83T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 - 12.4iT - 97T^{2} \)
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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

−9.381206685658348352301697771949, −8.677210525629967619578701490213, −7.76786657938386392465521164768, −7.39468889050557026902291444415, −6.14194779357999031084192495117, −5.16399698828556954473257136145, −3.60475489383521346940886552595, −3.33672065753020288777987859850, −1.79024571694650763305685482426, −0.55370687474428030260207714205, 1.50765809878986385831296269855, 2.31159566732134880403207509600, 4.31705966037614199373596755581, 5.04887455282325269997835255730, 5.76049295867320916663030440787, 7.04183621995883233249140762691, 7.41354939706118479578494373375, 8.407989631386667806573274827584, 9.218014373088256773252077714762, 9.582278595009961120785281172720

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