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Properties

Label 2-1248-39.35-c0-0-1
Degree $2$
Conductor $1248$
Sign $0.869 - 0.494i$
Analytic cond. $0.622833$
Root an. cond. $0.789197$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)3-s + i·5-s + (0.707 + 1.22i)7-s + (0.866 − 0.499i)9-s + (−1.22 − 0.707i)11-s + (0.5 − 0.866i)13-s + (0.258 + 0.965i)15-s + (−0.866 + 0.5i)17-s + (1 + 0.999i)21-s + (0.707 − 0.707i)27-s + (0.866 + 0.5i)29-s − 1.41·31-s + (−1.36 − 0.366i)33-s + (−1.22 + 0.707i)35-s + (−0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)3-s + i·5-s + (0.707 + 1.22i)7-s + (0.866 − 0.499i)9-s + (−1.22 − 0.707i)11-s + (0.5 − 0.866i)13-s + (0.258 + 0.965i)15-s + (−0.866 + 0.5i)17-s + (1 + 0.999i)21-s + (0.707 − 0.707i)27-s + (0.866 + 0.5i)29-s − 1.41·31-s + (−1.36 − 0.366i)33-s + (−1.22 + 0.707i)35-s + (−0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :0),\ 0.869 - 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.499151875\)
\(L(\frac12)\) \(\approx\) \(1.499151875\)
\(L(1)\) 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 \( 1 + (-0.965 + 0.258i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 - iT - T^{2} \)
7 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−10.02313558393422197528781108822, −8.757764119539186606571599775961, −8.470045911418286728618948307619, −7.70722563692914368694142078066, −6.75346633663972020246785291154, −5.83751223132998090947862696697, −4.94331673497049803118806441458, −3.44323029011350608793954721525, −2.80183056313832240161191472221, −1.93757189278324880619761544474, 1.41576279146415904955649125061, 2.50235074310734571661476950077, 4.02087760884868305983525208963, 4.50196554131274759940503401264, 5.23859232873305156820574146131, 6.87262181630164356816789815802, 7.55423881548259952116864087837, 8.222808022760686758073967816785, 9.007964435724423960803062527754, 9.647794750459414932928416496053

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