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Properties

Label 2-1260-7.4-c1-0-3
Degree $2$
Conductor $1260$
Sign $0.386 - 0.922i$
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  + (−0.5 − 0.866i)5-s + (−2.5 − 0.866i)7-s + (−2 + 3.46i)11-s + 7·13-s + (−3 + 5.19i)17-s + (−1.5 − 2.59i)19-s + (−1 − 1.73i)23-s + (−0.499 + 0.866i)25-s + 2·29-s + (−3.5 + 6.06i)31-s + (0.500 + 2.59i)35-s + (3.5 + 6.06i)37-s + 8·41-s + 5·43-s + (5 + 8.66i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.944 − 0.327i)7-s + (−0.603 + 1.04i)11-s + 1.94·13-s + (−0.727 + 1.26i)17-s + (−0.344 − 0.596i)19-s + (−0.208 − 0.361i)23-s + (−0.0999 + 0.173i)25-s + 0.371·29-s + (−0.628 + 1.08i)31-s + (0.0845 + 0.439i)35-s + (0.575 + 0.996i)37-s + 1.24·41-s + 0.762·43-s + (0.729 + 1.26i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.123385055\)
\(L(\frac12)\) \(\approx\) \(1.123385055\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 7T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-5 - 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4 - 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (7.5 - 12.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.854841928857544441656671985006, −8.915855268582601702154142891928, −8.379815792294676150069973695893, −7.34947147632109602011473184340, −6.46413072487416308669651032960, −5.84890174401273701277718597407, −4.47587220652628025851132830824, −3.90886557178600735210015007608, −2.69372013546873908501875540809, −1.25171298357288026216613093534, 0.52099052843588142411081167438, 2.39985290215922201020112336055, 3.37205058030186558840454043625, 4.09285998962856956311189882601, 5.70351554180458461424543216130, 6.03141849305104832464210902560, 7.01161671093974062175757734310, 7.971589575672985680435574219341, 8.800821680521949118612347702871, 9.394532182480944874267188999577

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