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Properties

Label 2-1260-12.11-c1-0-16
Degree $2$
Conductor $1260$
Sign $0.999 - 0.0214i$
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  + (−1.26 − 0.636i)2-s + (1.18 + 1.60i)4-s − i·5-s − i·7-s + (−0.478 − 2.78i)8-s + (−0.636 + 1.26i)10-s − 2.48·11-s + 5.41·13-s + (−0.636 + 1.26i)14-s + (−1.17 + 3.82i)16-s + 7.95i·17-s + 6.96i·19-s + (1.60 − 1.18i)20-s + (3.14 + 1.58i)22-s + 0.834·23-s + ⋯
L(s)  = 1  + (−0.892 − 0.450i)2-s + (0.594 + 0.803i)4-s − 0.447i·5-s − 0.377i·7-s + (−0.169 − 0.985i)8-s + (−0.201 + 0.399i)10-s − 0.749·11-s + 1.50·13-s + (−0.170 + 0.337i)14-s + (−0.292 + 0.956i)16-s + 1.92i·17-s + 1.59i·19-s + (0.359 − 0.265i)20-s + (0.669 + 0.337i)22-s + 0.174·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0214i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.999 - 0.0214i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.038641089\)
\(L(\frac12)\) \(\approx\) \(1.038641089\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.636i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + iT \)
good11 \( 1 + 2.48T + 11T^{2} \)
13 \( 1 - 5.41T + 13T^{2} \)
17 \( 1 - 7.95iT - 17T^{2} \)
19 \( 1 - 6.96iT - 19T^{2} \)
23 \( 1 - 0.834T + 23T^{2} \)
29 \( 1 + 7.43iT - 29T^{2} \)
31 \( 1 + 1.06iT - 31T^{2} \)
37 \( 1 - 0.204T + 37T^{2} \)
41 \( 1 + 1.84iT - 41T^{2} \)
43 \( 1 - 4.38iT - 43T^{2} \)
47 \( 1 - 7.00T + 47T^{2} \)
53 \( 1 - 4.95iT - 53T^{2} \)
59 \( 1 - 4.06T + 59T^{2} \)
61 \( 1 - 4.54T + 61T^{2} \)
67 \( 1 - 9.50iT - 67T^{2} \)
71 \( 1 - 9.40T + 71T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 + 17.6iT - 79T^{2} \)
83 \( 1 + 8.48T + 83T^{2} \)
89 \( 1 - 13.7iT - 89T^{2} \)
97 \( 1 - 4.91T + 97T^{2} \)
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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

−9.836339990730458394102447744803, −8.734046848353287756708514058751, −8.187204496278014537799883224227, −7.68202147334387384220755910000, −6.33051335236820047136501133585, −5.79351981017762139165670081853, −4.10700525391982992311909413498, −3.60406154836015379281135128634, −2.08589146123127718241819510698, −1.07012917006628754725209693461, 0.72814076152258072831178651125, 2.33247967363459232097178891939, 3.20162219316183949588019416138, 4.93270305062817094804428974135, 5.56103249992668457153208176143, 6.73354863475202549761302540475, 7.10590822467295398871033455473, 8.156336222974331436110200298529, 8.915118775702998972832293092733, 9.453681611581219455884024390373

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