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Properties

Label 2-2340-260.167-c0-0-1
Degree $2$
Conductor $2340$
Sign $0.0685 + 0.997i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999i·8-s − 0.999i·10-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.133 + 0.5i)17-s + (−0.499 − 0.866i)20-s + (−0.499 − 0.866i)25-s + (0.866 + 0.499i)26-s + (0.866 − 0.5i)29-s + (−0.866 − 0.499i)32-s + (0.366 + 0.366i)34-s + (−0.866 − 1.5i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999i·8-s − 0.999i·10-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.133 + 0.5i)17-s + (−0.499 − 0.866i)20-s + (−0.499 − 0.866i)25-s + (0.866 + 0.499i)26-s + (0.866 − 0.5i)29-s + (−0.866 − 0.499i)32-s + (0.366 + 0.366i)34-s + (−0.866 − 1.5i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0685 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0685 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.0685 + 0.997i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ 0.0685 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.169520435\)
\(L(\frac12)\) \(\approx\) \(2.169520435\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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

−9.096713701726011420807575555466, −8.402583155664788994602536351827, −7.29346605852254943832861068955, −6.27738383091008390513079338350, −5.86930186113270934868107254434, −4.81256849062531741367931286955, −4.31040739339857216659515147712, −3.29222078788837956199935565656, −2.10459441704836291819434505634, −1.26711099409722401725594078406, 1.87612162852862824683263168521, 3.06412986090775477133388049195, 3.45216462216322413216569623716, 4.79063526288280567473940927309, 5.43228193840309335966466086369, 6.32959598610290596765816915725, 6.78983634175255012629334002550, 7.67767995667911654500272758827, 8.342394787758079696265290519007, 9.292439106892323469345638685892

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