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Properties

Label 2-1260-105.53-c1-0-4
Degree $2$
Conductor $1260$
Sign $0.416 - 0.909i$
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  + (2.03 + 0.918i)5-s + (1.31 + 2.29i)7-s + (5.56 + 3.21i)11-s + (1.53 + 1.53i)13-s + (−3.67 + 0.985i)17-s + (−1.19 + 0.687i)19-s + (−8.15 − 2.18i)23-s + (3.31 + 3.74i)25-s − 5.95·29-s + (4.97 − 8.61i)31-s + (0.566 + 5.88i)35-s + (−2.39 − 0.642i)37-s − 1.18i·41-s + (3.28 + 3.28i)43-s + (0.655 − 2.44i)47-s + ⋯
L(s)  = 1  + (0.911 + 0.410i)5-s + (0.495 + 0.868i)7-s + (1.67 + 0.967i)11-s + (0.425 + 0.425i)13-s + (−0.892 + 0.239i)17-s + (−0.273 + 0.157i)19-s + (−1.70 − 0.455i)23-s + (0.662 + 0.748i)25-s − 1.10·29-s + (0.893 − 1.54i)31-s + (0.0957 + 0.995i)35-s + (−0.394 − 0.105i)37-s − 0.184i·41-s + (0.500 + 0.500i)43-s + (0.0955 − 0.356i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.416 - 0.909i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.416 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.141866381\)
\(L(\frac12)\) \(\approx\) \(2.141866381\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.03 - 0.918i)T \)
7 \( 1 + (-1.31 - 2.29i)T \)
good11 \( 1 + (-5.56 - 3.21i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.53 - 1.53i)T + 13iT^{2} \)
17 \( 1 + (3.67 - 0.985i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.19 - 0.687i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (8.15 + 2.18i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.95T + 29T^{2} \)
31 \( 1 + (-4.97 + 8.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.39 + 0.642i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.18iT - 41T^{2} \)
43 \( 1 + (-3.28 - 3.28i)T + 43iT^{2} \)
47 \( 1 + (-0.655 + 2.44i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.482 - 1.80i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.27 - 5.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.09 - 7.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.49 + 9.29i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.06iT - 71T^{2} \)
73 \( 1 + (-12.9 + 3.47i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.00 + 4.62i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.11 - 5.11i)T - 83iT^{2} \)
89 \( 1 + (3.97 + 6.88i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.537 - 0.537i)T - 97iT^{2} \)
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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

−9.596451752232788662192724785924, −9.220180414845295120566320617144, −8.376057602503914177684222852908, −7.26804963625991852504024090296, −6.20670438666246053804474802885, −6.05656609883075790730198934836, −4.62406763801639192182776396284, −3.88124461055396629407306025258, −2.22859278147549796168496882245, −1.77153501487396043615819465719, 0.963237080099873913766715364469, 1.93601786121330708314766830305, 3.53739981884527001171502870724, 4.29585710163474862662965233772, 5.38060776698330216658659287151, 6.28448149047572426527020864697, 6.85268728161917893595601662092, 8.123829455695954972989876567424, 8.745177154022593472555823342607, 9.491773569529085739838779941667

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