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Properties

Label 2-2368-148.63-c0-0-3
Degree $2$
Conductor $2368$
Sign $0.989 + 0.146i$
Analytic cond. $1.18178$
Root an. cond. $1.08709$
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  + (1.22 + 0.707i)3-s + (0.5 − 0.866i)5-s + (0.499 + 0.866i)9-s − 1.41i·11-s + (1.22 − 0.707i)15-s + (0.5 + 0.866i)17-s + (−1.22 − 0.707i)19-s − 1.41i·23-s + 29-s + 1.41i·31-s + (1.00 − 1.73i)33-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + 0.999·45-s + 1.41i·47-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)3-s + (0.5 − 0.866i)5-s + (0.499 + 0.866i)9-s − 1.41i·11-s + (1.22 − 0.707i)15-s + (0.5 + 0.866i)17-s + (−1.22 − 0.707i)19-s − 1.41i·23-s + 29-s + 1.41i·31-s + (1.00 − 1.73i)33-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + 0.999·45-s + 1.41i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.989 + 0.146i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ 0.989 + 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.931350815\)
\(L(\frac12)\) \(\approx\) \(1.931350815\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−8.834839270715694422647189387048, −8.491958773108976338858247478434, −8.258398605746880742067083545445, −6.73778425216265687250778380004, −6.02136212758902710679559779629, −4.96357803476285424822897127234, −4.32230215560585224826499033512, −3.30804303683075415317614434879, −2.63833797010114967808489291620, −1.30086374534831050045268627494, 1.79416420341855152139976670330, 2.29363423656328928236118601786, 3.20284580988404302438291146336, 4.13469063478951122970755448270, 5.30832071683210329752197450541, 6.35090332261563620596099769196, 7.09979513386768516278040449988, 7.54510432041385767522655881762, 8.302158099468457275761385126311, 9.191997026715712042363511777506

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