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Properties

Label 2-2340-780.407-c0-0-3
Degree $2$
Conductor $2340$
Sign $0.671 + 0.740i$
Analytic cond. $1.16781$
Root an. cond. $1.08065$
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)2-s + (0.866 − 0.499i)4-s + (0.965 − 0.258i)5-s + (0.707 − 0.707i)8-s + (0.866 − 0.499i)10-s + (−0.866 − 0.5i)13-s + (0.500 − 0.866i)16-s + (−0.965 − 0.258i)17-s + (0.707 − 0.707i)20-s + (0.866 − 0.499i)25-s + (−0.965 − 0.258i)26-s + (−0.258 + 0.448i)29-s + (0.258 − 0.965i)32-s − 34-s + (0.5 + 1.86i)37-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.965 − 0.258i)5-s + (0.707 − 0.707i)8-s + (0.866 − 0.499i)10-s + (−0.866 − 0.5i)13-s + (0.500 − 0.866i)16-s + (−0.965 − 0.258i)17-s + (0.707 − 0.707i)20-s + (0.866 − 0.499i)25-s + (−0.965 − 0.258i)26-s + (−0.258 + 0.448i)29-s + (0.258 − 0.965i)32-s − 34-s + (0.5 + 1.86i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ 0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.462178695\)
\(L(\frac12)\) \(\approx\) \(2.462178695\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-0.965 + 0.258i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.266160589923601456523211934631, −8.300376580819603975089594757582, −7.26756505825088980668967175697, −6.53932437670378227096962824155, −5.85929980902519358887031026086, −4.95267706198385216558014496432, −4.54251606741308614482999287811, −3.15709764164150563705942974626, −2.45378368522655505216418956643, −1.40436854176327960303116178774, 1.96519854175137215162007772210, 2.47298462530004786320268287053, 3.70116110755786944808710433283, 4.55672987307466428271945642699, 5.41689766433682559216525610710, 6.03565988749367902015122391696, 6.92010203152529809646580607005, 7.32862131459865771201521953720, 8.499057843973269879029067294755, 9.260071747058135178125317424021

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