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Properties

Label 2-2340-39.2-c1-0-4
Degree $2$
Conductor $2340$
Sign $0.254 - 0.967i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
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  + (−0.707 − 0.707i)5-s + (2.64 − 0.709i)7-s + (2.92 + 0.782i)11-s + (−1.99 + 3.00i)13-s + (−3.98 + 6.89i)17-s + (0.857 + 3.20i)19-s + (−1.87 − 3.24i)23-s + 1.00i·25-s + (−4.44 + 2.56i)29-s + (3.36 − 3.36i)31-s + (−2.37 − 1.36i)35-s + (−1.38 + 5.15i)37-s + (−1.33 + 4.99i)41-s + (6.51 + 3.76i)43-s + (−6.12 + 6.12i)47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s + (1.00 − 0.268i)7-s + (0.880 + 0.236i)11-s + (−0.552 + 0.833i)13-s + (−0.966 + 1.67i)17-s + (0.196 + 0.734i)19-s + (−0.390 − 0.675i)23-s + 0.200i·25-s + (−0.825 + 0.476i)29-s + (0.604 − 0.604i)31-s + (−0.401 − 0.231i)35-s + (−0.226 + 0.846i)37-s + (−0.208 + 0.779i)41-s + (0.994 + 0.574i)43-s + (−0.893 + 0.893i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.254 - 0.967i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.254 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.526381089\)
\(L(\frac12)\) \(\approx\) \(1.526381089\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (1.99 - 3.00i)T \)
good7 \( 1 + (-2.64 + 0.709i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.92 - 0.782i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.98 - 6.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.857 - 3.20i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.87 + 3.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.44 - 2.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.36 + 3.36i)T - 31iT^{2} \)
37 \( 1 + (1.38 - 5.15i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.33 - 4.99i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.51 - 3.76i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.12 - 6.12i)T - 47iT^{2} \)
53 \( 1 + 7.31iT - 53T^{2} \)
59 \( 1 + (0.977 + 3.64i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.79 + 4.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.13 + 1.10i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.64 - 0.977i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-6.65 - 6.65i)T + 73iT^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + (-5.35 - 5.35i)T + 83iT^{2} \)
89 \( 1 + (-13.1 - 3.52i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.09 - 4.07i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.126746203784946430062648340736, −8.178189198356556541152103949412, −7.913111078227661338525615541075, −6.71169799514912225112859570232, −6.22666695536465416141647744653, −4.95246555529045592597667597523, −4.34100762004361331287337433590, −3.71431610484278314899006388179, −2.09338675392223656129142911310, −1.38081575509345846282928075955, 0.53149444365813857288792584473, 2.01414513106529546361699811929, 2.93277901701215310232880058323, 4.00605094753597288217959561191, 4.91111785903200098541336185867, 5.52030532938487828226756370585, 6.65670771942870103584923731313, 7.36315195048218352754837507175, 7.932009795209857071361024666157, 8.999270497469127280140529574191

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