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Properties

Label 2-1260-315.139-c0-0-1
Degree $2$
Conductor $1260$
Sign $0.766 - 0.642i$
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 + (0.5 + 0.866i)11-s + (1 − 1.73i)13-s + 0.999·15-s − 2·17-s + (−0.499 + 0.866i)21-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)33-s + 0.999·35-s + 1.99·39-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 + (0.5 + 0.866i)11-s + (1 − 1.73i)13-s + 0.999·15-s − 2·17-s + (−0.499 + 0.866i)21-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)33-s + 0.999·35-s + 1.99·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.766 - 0.642i$
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: \((2,\ 1260,\ (\ :0),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.416075455\)
\(L(\frac12)\) \(\approx\) \(1.416075455\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 2T + T^{2} \)
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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.864469230673860782935480899690, −8.950824591696030781574966156210, −8.668079352101774891437031947705, −7.927287338321563240811954957928, −6.51042018249083745981174086560, −5.50908495733405402775612947779, −4.90388472246914989302735137791, −4.06775105549190429450854116228, −2.78001625726864448268340408729, −1.75699795086470092166979045947, 1.45994906823908556845935474004, 2.36820505573428217124857433547, 3.63311597893621689150997821260, 4.40002034353953544830578766545, 6.13313138525700253975617482751, 6.55572790767979723827497821706, 7.12597916282249765740688335300, 8.220780031561802450934115196483, 8.901788031170369103447288862333, 9.594761034038477651683029524446

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