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Properties

Label 2-1260-35.4-c1-0-4
Degree $2$
Conductor $1260$
Sign $-0.866 - 0.499i$
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  + (0.403 + 2.19i)5-s + (−0.308 + 2.62i)7-s + (−0.919 + 1.59i)11-s + 0.356i·13-s + (−3.16 − 1.82i)17-s + (0.936 + 1.62i)19-s + (−5.76 + 3.33i)23-s + (−4.67 + 1.77i)25-s + 2.37·29-s + (3.37 − 5.84i)31-s + (−5.90 + 0.380i)35-s + (3.66 − 2.11i)37-s − 1.83·41-s − 2.80i·43-s + (−7.63 + 4.40i)47-s + ⋯
L(s)  = 1  + (0.180 + 0.983i)5-s + (−0.116 + 0.993i)7-s + (−0.277 + 0.480i)11-s + 0.0988i·13-s + (−0.768 − 0.443i)17-s + (0.214 + 0.372i)19-s + (−1.20 + 0.694i)23-s + (−0.935 + 0.354i)25-s + 0.440·29-s + (0.605 − 1.04i)31-s + (−0.997 + 0.0642i)35-s + (0.602 − 0.347i)37-s − 0.287·41-s − 0.427i·43-s + (−1.11 + 0.643i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.866 - 0.499i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.866 - 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.036668120\)
\(L(\frac12)\) \(\approx\) \(1.036668120\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.403 - 2.19i)T \)
7 \( 1 + (0.308 - 2.62i)T \)
good11 \( 1 + (0.919 - 1.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.356iT - 13T^{2} \)
17 \( 1 + (3.16 + 1.82i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.936 - 1.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.76 - 3.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.37T + 29T^{2} \)
31 \( 1 + (-3.37 + 5.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.66 + 2.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.83T + 41T^{2} \)
43 \( 1 + 2.80iT - 43T^{2} \)
47 \( 1 + (7.63 - 4.40i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.93 + 5.15i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.86 - 8.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.436 + 0.756i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.0 - 5.78i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + (7.90 + 4.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.93 - 5.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.66iT - 83T^{2} \)
89 \( 1 + (-3.29 - 5.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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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

−9.760798761714611065466068291907, −9.551200469888614858329629305748, −8.296478936284156396989237704132, −7.62157244288093013940900998805, −6.59225484131209039671085632830, −6.01161718613755969389181345096, −5.04446103170968822006482769327, −3.87079343974376345933038883841, −2.73603914167999500783575832528, −2.00789522609397022426011194578, 0.41536277540471189312704264351, 1.71237892758716889292619636917, 3.18138663971701237650265688166, 4.32770102989324705515111391578, 4.89559033931789924242606602396, 6.08691315802360750460896437837, 6.76418447162933408629938149512, 8.014582916190027909436341686514, 8.357596790901561338607033531737, 9.399301049135942855017771904756

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