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Properties

Label 2-1148-7.4-c1-0-25
Degree $2$
Conductor $1148$
Sign $-0.989 + 0.143i$
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  + (1.71 − 2.96i)3-s + (−1.17 − 2.04i)5-s + (1.97 − 1.76i)7-s + (−4.36 − 7.55i)9-s + (0.0714 − 0.123i)11-s − 2.89·13-s − 8.07·15-s + (0.697 − 1.20i)17-s + (3.61 + 6.26i)19-s + (−1.86 − 8.86i)21-s + (2.35 + 4.07i)23-s + (−0.278 + 0.482i)25-s − 19.5·27-s + 2.50·29-s + (1.14 − 1.99i)31-s + ⋯
L(s)  = 1  + (0.988 − 1.71i)3-s + (−0.527 − 0.913i)5-s + (0.744 − 0.667i)7-s + (−1.45 − 2.51i)9-s + (0.0215 − 0.0373i)11-s − 0.801·13-s − 2.08·15-s + (0.169 − 0.292i)17-s + (0.829 + 1.43i)19-s + (−0.406 − 1.93i)21-s + (0.490 + 0.848i)23-s + (−0.0557 + 0.0965i)25-s − 3.77·27-s + 0.464·29-s + (0.206 − 0.357i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.989 + 0.143i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.989 + 0.143i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.050298328\)
\(L(\frac12)\) \(\approx\) \(2.050298328\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.97 + 1.76i)T \)
41 \( 1 + T \)
good3 \( 1 + (-1.71 + 2.96i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.17 + 2.04i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.0714 + 0.123i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.89T + 13T^{2} \)
17 \( 1 + (-0.697 + 1.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.61 - 6.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.35 - 4.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.50T + 29T^{2} \)
31 \( 1 + (-1.14 + 1.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.47 - 7.75i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + (-0.288 - 0.500i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.33 + 2.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.50 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.09 + 8.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.40 - 4.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (-1.03 + 1.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.37 + 2.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.58T + 83T^{2} \)
89 \( 1 + (5.67 + 9.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.73T + 97T^{2} \)
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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

−9.123341743218621998652019357846, −8.277260341843925613753543434320, −7.72614255320423138185186909109, −7.38119699180698293196941397129, −6.26602089203864915770151626640, −5.13710985967072315770281332271, −3.96417759532065703097830965018, −2.90056793152823433253402193426, −1.60904040744632673820078427269, −0.834957464892811042806548899266, 2.57542609675241847297412976008, 2.88377815730299217523515298651, 4.13389760539492010969332406833, 4.79144488829986961220208979731, 5.62767073497069190705671719123, 7.20720075057368948781407093391, 7.85549464984255338238226303330, 8.840908323594607755101717936405, 9.232717622051721770395652478353, 10.19793940879373989103570580569

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