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Properties

Label 2-1260-315.34-c0-0-0
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: $\chi_{1260} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4133396438\)
\(L(\frac12)\) \(\approx\) \(0.4133396438\)
\(L(1)\) 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

−9.960404819097634586826001883482, −9.161810921855768919372592667284, −8.369207584681169553210032516886, −7.60352544086828354206564663855, −6.78640693361545804290467268543, −5.97098949285781605426479310695, −4.95586912477766349241103116159, −4.35778895576497876849218120891, −2.60576088608570906903360334815, −1.71354733364478025856655651242, 0.36539991274136573843907421278, 3.05916145408168353942793336503, 3.38628760539103877969484622366, 4.45849541859484611406644373857, 5.58986539697941753211106472356, 6.29577245621593577565232758370, 7.12834963442663261870102134463, 8.209836644865922031817893005036, 8.813447934972045669805243853457, 10.17571133057321982192217242226

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