Properties

Label 2-1148-7.2-c1-0-21
Degree $2$
Conductor $1148$
Sign $0.676 + 0.736i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
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  + (1.14 + 1.98i)3-s + (1.96 − 3.39i)5-s + (−0.412 − 2.61i)7-s + (−1.12 + 1.95i)9-s + (−1.75 − 3.04i)11-s − 3.61·13-s + 8.99·15-s + (3.34 + 5.79i)17-s + (−0.103 + 0.179i)19-s + (4.71 − 3.81i)21-s + (1.39 − 2.41i)23-s + (−5.19 − 8.99i)25-s + 1.71·27-s − 2.98·29-s + (−2.14 − 3.72i)31-s + ⋯
L(s)  = 1  + (0.661 + 1.14i)3-s + (0.877 − 1.51i)5-s + (−0.155 − 0.987i)7-s + (−0.375 + 0.650i)9-s + (−0.529 − 0.917i)11-s − 1.00·13-s + 2.32·15-s + (0.811 + 1.40i)17-s + (−0.0237 + 0.0411i)19-s + (1.02 − 0.831i)21-s + (0.290 − 0.503i)23-s + (−1.03 − 1.79i)25-s + 0.329·27-s − 0.554·29-s + (−0.385 − 0.668i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.676 + 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.124390905\)
\(L(\frac12)\) \(\approx\) \(2.124390905\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.412 + 2.61i)T \)
41 \( 1 + T \)
good3 \( 1 + (-1.14 - 1.98i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.96 + 3.39i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.75 + 3.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.61T + 13T^{2} \)
17 \( 1 + (-3.34 - 5.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.103 - 0.179i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.39 + 2.41i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.98T + 29T^{2} \)
31 \( 1 + (2.14 + 3.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.14 + 8.90i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 9.05T + 43T^{2} \)
47 \( 1 + (-5.26 + 9.11i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.35 - 11.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.06 + 7.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.90 - 3.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.29 - 3.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.96T + 71T^{2} \)
73 \( 1 + (-7.03 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.17 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.55T + 83T^{2} \)
89 \( 1 + (1.16 - 2.01i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.84T + 97T^{2} \)
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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

−9.592798536404643323174553796404, −9.063115771763597167155065562051, −8.303179727081859297282964426553, −7.50904176847298913314402697888, −5.95998019420261553889579886120, −5.33184202203623319586275024858, −4.33873717604317174077448412848, −3.76736781739050535025531133299, −2.38869217462589660625868104975, −0.850819167544585156853834300589, 1.80463087387018941950716158119, 2.67992848508283382107893024725, 2.94358408335689166811283006557, 5.00667812769764269348778982844, 5.86330338581860429457911060283, 6.88505450243839352250193228222, 7.28651727854691024152396733555, 7.958059444572824494364871131441, 9.356379454750020593128732065491, 9.667979739303150157015816585041

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