Properties

Label 2-1260-7.4-c1-0-13
Degree $2$
Conductor $1260$
Sign $-0.895 + 0.444i$
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 + (−1.32 − 2.29i)7-s + (−0.822 + 1.42i)11-s − 2.64·13-s + (−0.822 + 1.42i)17-s + (−4.14 − 7.18i)19-s + (0.822 + 1.42i)23-s + (−0.499 + 0.866i)25-s − 7.64·29-s + (2.14 − 3.71i)31-s + (1.32 − 2.29i)35-s + (−0.322 − 0.559i)37-s − 4.93·41-s − 5.93·43-s + (3 + 5.19i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.499 − 0.866i)7-s + (−0.248 + 0.429i)11-s − 0.733·13-s + (−0.199 + 0.345i)17-s + (−0.951 − 1.64i)19-s + (0.171 + 0.297i)23-s + (−0.0999 + 0.173i)25-s − 1.41·29-s + (0.385 − 0.667i)31-s + (0.223 − 0.387i)35-s + (−0.0530 − 0.0919i)37-s − 0.771·41-s − 0.905·43-s + (0.437 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.895 + 0.444i$
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: \((2,\ 1260,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3820639578\)
\(L(\frac12)\) \(\approx\) \(0.3820639578\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 2.29i)T \)
good11 \( 1 + (0.822 - 1.42i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
17 \( 1 + (0.822 - 1.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.14 + 7.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.822 - 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.64T + 29T^{2} \)
31 \( 1 + (-2.14 + 3.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.322 + 0.559i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.93T + 41T^{2} \)
43 \( 1 + 5.93T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.64 + 2.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.46 + 9.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.322 - 0.559i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + (6.61 - 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.14 + 1.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (7.11 + 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−9.543425911724097622563503855303, −8.579592519264673130177676569409, −7.44671282780646900972383631867, −6.99055892359695165805950821657, −6.15674738943037634945185639912, −4.98300139744358199608984380729, −4.16038531675518489587041313879, −3.04959555633982764802269314723, −1.99080297305571502335173203087, −0.14719914391265590872760251820, 1.79725246722893934108647973852, 2.84095122546039708125001086604, 3.95871108753831080156254824841, 5.13700009680785864729033428788, 5.79417144096410689711830311289, 6.61889529485795986571443313741, 7.67852864705903701400670717659, 8.578469330730447818193935983751, 9.108673003345525662376340415421, 10.05664713180065520842083800461

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