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Properties

Label 2-1260-315.139-c0-0-3
Degree $2$
Conductor $1260$
Sign $0.173 + 0.984i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
Motivic weight $0$
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)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)13-s + (−0.499 − 0.866i)15-s + 17-s + 0.999·21-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (0.5 + 0.866i)29-s − 1.99·33-s + 0.999·35-s + (0.499 + 0.866i)39-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)13-s + (−0.499 − 0.866i)15-s + 17-s + 0.999·21-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (0.5 + 0.866i)29-s − 1.99·33-s + 0.999·35-s + (0.499 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.323617775\)
\(L(\frac12)\) \(\approx\) \(1.323617775\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.324586641205024061316566835828, −8.811614332750305123739127080029, −8.189066307730172023262959205682, −7.50535857912602270935941050567, −6.15719941987932410935563736091, −5.68064126024438278805083799786, −4.78350001836303795245870347757, −3.23114226434949673716754199553, −2.35474041582339098121815082069, −1.18824253690822879421096817828, 2.07613750332002223419713493323, 2.92182032578650271232824187533, 4.02143501024977787748642665702, 4.94135775164206936266334076984, 5.62734940779698899976639609274, 7.12475586775523253315733268132, 7.56326752910689527929695213211, 8.320959207229377720083476956294, 9.721447906215663197985108106235, 10.10800683484213971464392342760

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