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Properties

Label 2-1260-35.9-c1-0-11
Degree $2$
Conductor $1260$
Sign $0.971 + 0.238i$
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  + (−2.13 + 0.656i)5-s + (1.13 + 2.38i)7-s + (−2.63 − 4.56i)11-s − 2.62i·13-s + (−0.362 + 0.209i)17-s + (1.63 − 2.83i)19-s + (6.77 + 3.91i)23-s + (4.13 − 2.80i)25-s + 4.27·29-s + (−1.63 − 2.83i)31-s + (−4 − 4.35i)35-s + (8.63 + 4.98i)37-s + 3.72·41-s + 2.15i·43-s + (5.63 + 3.25i)47-s + ⋯
L(s)  = 1  + (−0.955 + 0.293i)5-s + (0.429 + 0.902i)7-s + (−0.795 − 1.37i)11-s − 0.728i·13-s + (−0.0879 + 0.0507i)17-s + (0.375 − 0.650i)19-s + (1.41 + 0.815i)23-s + (0.827 − 0.561i)25-s + 0.793·29-s + (−0.294 − 0.509i)31-s + (−0.676 − 0.736i)35-s + (1.41 + 0.819i)37-s + 0.581·41-s + 0.327i·43-s + (0.822 + 0.474i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.971 + 0.238i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.971 + 0.238i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.341108617\)
\(L(\frac12)\) \(\approx\) \(1.341108617\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 - 0.656i)T \)
7 \( 1 + (-1.13 - 2.38i)T \)
good11 \( 1 + (2.63 + 4.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.62iT - 13T^{2} \)
17 \( 1 + (0.362 - 0.209i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.63 + 2.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.77 - 3.91i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 + (1.63 + 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.63 - 4.98i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.72T + 41T^{2} \)
43 \( 1 - 2.15iT - 43T^{2} \)
47 \( 1 + (-5.63 - 3.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.91 - 2.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.63 - 2.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.77 + 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.04 + 1.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.54T + 71T^{2} \)
73 \( 1 + (5.63 - 3.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.63 + 6.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.40iT - 83T^{2} \)
89 \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−9.537622529356355252472763923224, −8.668682954253171421567701503225, −8.060797251144186910784856588637, −7.43231684460042916142243478426, −6.24260177321851879161291704315, −5.42424324855726557040754554299, −4.61719668959566850266000957200, −3.20873576837898858020654706755, −2.75278272432819446987660799332, −0.76757521080139810802170603735, 0.987652687564147216651665321081, 2.44951994673837380359714826067, 3.83933341682787114833953574231, 4.54020982356149166871691173342, 5.18444937194707060705454569763, 6.75374886475519357756019139143, 7.32338072138089941322510743753, 7.943743769735318628797315722544, 8.835218351821493536117642934699, 9.764081610501987044782989442616

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