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Properties

Label 2-1260-105.2-c1-0-13
Degree $2$
Conductor $1260$
Sign $0.516 + 0.856i$
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  + (−0.260 − 2.22i)5-s + (2.63 + 0.260i)7-s + (−3.34 + 1.93i)11-s + (4.35 − 4.35i)13-s + (4.66 + 1.24i)17-s + (1.57 + 0.907i)19-s + (−2.15 + 0.576i)23-s + (−4.86 + 1.15i)25-s + 0.529·29-s + (−3.19 − 5.53i)31-s + (−0.106 − 5.91i)35-s + (9.69 − 2.59i)37-s − 5.31i·41-s + (−1.82 + 1.82i)43-s + (−2.59 − 9.69i)47-s + ⋯
L(s)  = 1  + (−0.116 − 0.993i)5-s + (0.995 + 0.0985i)7-s + (−1.00 + 0.581i)11-s + (1.20 − 1.20i)13-s + (1.13 + 0.302i)17-s + (0.360 + 0.208i)19-s + (−0.448 + 0.120i)23-s + (−0.972 + 0.231i)25-s + 0.0983·29-s + (−0.573 − 0.993i)31-s + (−0.0179 − 0.999i)35-s + (1.59 − 0.427i)37-s − 0.830i·41-s + (−0.277 + 0.277i)43-s + (−0.378 − 1.41i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.516 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.789328815\)
\(L(\frac12)\) \(\approx\) \(1.789328815\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.260 + 2.22i)T \)
7 \( 1 + (-2.63 - 0.260i)T \)
good11 \( 1 + (3.34 - 1.93i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.35 + 4.35i)T - 13iT^{2} \)
17 \( 1 + (-4.66 - 1.24i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.57 - 0.907i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.15 - 0.576i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.529T + 29T^{2} \)
31 \( 1 + (3.19 + 5.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.69 + 2.59i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.31iT - 41T^{2} \)
43 \( 1 + (1.82 - 1.82i)T - 43iT^{2} \)
47 \( 1 + (2.59 + 9.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.0180 + 0.0673i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.07 - 1.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.99 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.36 - 5.08i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.72iT - 71T^{2} \)
73 \( 1 + (3.55 + 0.953i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.564 + 0.325i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.54 - 4.54i)T + 83iT^{2} \)
89 \( 1 + (7.34 - 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.639 - 0.639i)T + 97iT^{2} \)
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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

−9.593181233021064074912375100405, −8.478959053529875964499419948311, −8.012380564241745599151047550955, −7.49798215578044715343072384035, −5.74127703773349599610428633852, −5.50995329260965415262464456329, −4.47645519376368157837641356050, −3.49163161905986153295114192567, −2.03277416584103877800775431872, −0.863422937145067253426833535321, 1.37058927993204504493813955551, 2.69150521869781622733096357517, 3.62681874767399227138070074664, 4.68556200831075450773852934820, 5.72106528101934282956138084745, 6.48549239000618830439769576766, 7.53822169699830589469066083216, 8.027115194437476059977800512799, 8.927940939547551647573973142703, 9.978974488614586983132370395649

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