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Properties

Label 2-1260-35.13-c1-0-15
Degree $2$
Conductor $1260$
Sign $0.494 + 0.869i$
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  + (2.22 − 0.248i)5-s + (1.51 − 2.16i)7-s − 1.10·11-s + (3.52 − 3.52i)13-s + (−3.23 − 3.23i)17-s + 0.916·19-s + (−4.43 − 4.43i)23-s + (4.87 − 1.10i)25-s + 8.28i·29-s − 4.59i·31-s + (2.83 − 5.19i)35-s + (−2.30 + 2.30i)37-s + 3.15i·41-s + (2.70 + 2.70i)43-s + (−4.71 − 4.71i)47-s + ⋯
L(s)  = 1  + (0.993 − 0.111i)5-s + (0.574 − 0.818i)7-s − 0.333·11-s + (0.977 − 0.977i)13-s + (−0.785 − 0.785i)17-s + 0.210·19-s + (−0.925 − 0.925i)23-s + (0.975 − 0.221i)25-s + 1.53i·29-s − 0.825i·31-s + (0.479 − 0.877i)35-s + (−0.378 + 0.378i)37-s + 0.492i·41-s + (0.412 + 0.412i)43-s + (−0.687 − 0.687i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.494 + 0.869i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.494 + 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.031789397\)
\(L(\frac12)\) \(\approx\) \(2.031789397\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 + 0.248i)T \)
7 \( 1 + (-1.51 + 2.16i)T \)
good11 \( 1 + 1.10T + 11T^{2} \)
13 \( 1 + (-3.52 + 3.52i)T - 13iT^{2} \)
17 \( 1 + (3.23 + 3.23i)T + 17iT^{2} \)
19 \( 1 - 0.916T + 19T^{2} \)
23 \( 1 + (4.43 + 4.43i)T + 23iT^{2} \)
29 \( 1 - 8.28iT - 29T^{2} \)
31 \( 1 + 4.59iT - 31T^{2} \)
37 \( 1 + (2.30 - 2.30i)T - 37iT^{2} \)
41 \( 1 - 3.15iT - 41T^{2} \)
43 \( 1 + (-2.70 - 2.70i)T + 43iT^{2} \)
47 \( 1 + (4.71 + 4.71i)T + 47iT^{2} \)
53 \( 1 + (-6.41 - 6.41i)T + 53iT^{2} \)
59 \( 1 - 4.36T + 59T^{2} \)
61 \( 1 + 10.8iT - 61T^{2} \)
67 \( 1 + (6.81 - 6.81i)T - 67iT^{2} \)
71 \( 1 + 4.79T + 71T^{2} \)
73 \( 1 + (-9.72 + 9.72i)T - 73iT^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 + (0.227 - 0.227i)T - 83iT^{2} \)
89 \( 1 - 8.93T + 89T^{2} \)
97 \( 1 + (-12.4 - 12.4i)T + 97iT^{2} \)
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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

−9.624770606653405498471178997956, −8.692611188200242870143360126694, −8.026877487016342460712346021816, −7.03196944705333103695920990053, −6.21226894109614244321519292839, −5.30382829577667080292007944009, −4.53426445259275758700619393140, −3.31383995947158872319875871758, −2.13356579479650656524938203560, −0.891672416044401442045924000903, 1.65693670130773245850349755315, 2.31339379714240164651414854623, 3.74130531519848885142473600165, 4.81036852522241907737188967624, 5.84948904793525339432959135368, 6.21329456213008790571371599629, 7.35341955953755668660438254221, 8.466569097726848797919005207272, 8.921050777222911260504196418860, 9.778576805642181252593532800794

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