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Properties

Label 2-1260-105.89-c1-0-8
Degree $2$
Conductor $1260$
Sign $0.993 + 0.113i$
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  + (2.03 + 0.930i)5-s + (−2.30 − 1.29i)7-s + (1.33 − 0.768i)11-s + 0.219·13-s + (1.23 − 0.713i)17-s + (3.80 + 2.19i)19-s + (0.389 − 0.673i)23-s + (3.26 + 3.78i)25-s − 6.82i·29-s + (5.32 − 3.07i)31-s + (−3.47 − 4.78i)35-s + (5.19 + 2.99i)37-s + 8.90·41-s + 1.91i·43-s + (−9.70 − 5.60i)47-s + ⋯
L(s)  = 1  + (0.909 + 0.416i)5-s + (−0.871 − 0.490i)7-s + (0.401 − 0.231i)11-s + 0.0608·13-s + (0.299 − 0.172i)17-s + (0.872 + 0.503i)19-s + (0.0811 − 0.140i)23-s + (0.653 + 0.756i)25-s − 1.26i·29-s + (0.956 − 0.552i)31-s + (−0.587 − 0.808i)35-s + (0.853 + 0.492i)37-s + 1.39·41-s + 0.292i·43-s + (−1.41 − 0.817i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.993 + 0.113i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.993 + 0.113i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.880086141\)
\(L(\frac12)\) \(\approx\) \(1.880086141\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.03 - 0.930i)T \)
7 \( 1 + (2.30 + 1.29i)T \)
good11 \( 1 + (-1.33 + 0.768i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.219T + 13T^{2} \)
17 \( 1 + (-1.23 + 0.713i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.80 - 2.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.389 + 0.673i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.82iT - 29T^{2} \)
31 \( 1 + (-5.32 + 3.07i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.19 - 2.99i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.90T + 41T^{2} \)
43 \( 1 - 1.91iT - 43T^{2} \)
47 \( 1 + (9.70 + 5.60i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.41 - 4.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.336 - 0.582i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.64 - 5.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.78 - 5.65i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + (-1.09 - 1.88i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.19 + 5.53i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.09iT - 83T^{2} \)
89 \( 1 + (-2.97 + 5.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−9.868340132715226056082151788578, −9.102868518122027011322864366999, −7.961680168145933549989180168724, −7.11849724164601607460002323306, −6.23360063553073772871856518531, −5.78340404403258633039104612200, −4.46719395472507158589050179279, −3.39759275771071785253352961165, −2.51922384683872953693459683819, −1.01982292106097575837759592992, 1.13222910047641101428011279028, 2.46463146166390815267938470721, 3.41616093697233787539808150730, 4.73616873663958515897348820955, 5.55781945734245717643442009781, 6.33565663239719434308563702072, 7.06251994046893184274779725994, 8.232793684545940712867397074589, 9.161713414668231600858395146892, 9.534437334080630989480985362723

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