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Properties

Label 2-2340-39.32-c1-0-17
Degree $2$
Conductor $2340$
Sign $-0.426 + 0.904i$
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 + (1.29 − 4.83i)7-s + (−1.26 − 4.72i)11-s + (3.20 − 1.65i)13-s + (−1.47 − 2.55i)17-s + (3.97 + 1.06i)19-s + (−0.451 + 0.782i)23-s + 1.00i·25-s + (−4.66 − 2.69i)29-s + (−4.05 + 4.05i)31-s + (4.33 − 2.50i)35-s + (−4.50 + 1.20i)37-s + (−6.53 + 1.74i)41-s + (5.10 − 2.94i)43-s + (−3.65 + 3.65i)47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + (0.489 − 1.82i)7-s + (−0.381 − 1.42i)11-s + (0.888 − 0.458i)13-s + (−0.357 − 0.620i)17-s + (0.912 + 0.244i)19-s + (−0.0941 + 0.163i)23-s + 0.200i·25-s + (−0.865 − 0.499i)29-s + (−0.727 + 0.727i)31-s + (0.732 − 0.422i)35-s + (−0.740 + 0.198i)37-s + (−1.01 + 0.273i)41-s + (0.778 − 0.449i)43-s + (−0.532 + 0.532i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.426 + 0.904i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.426 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.692770467\)
\(L(\frac12)\) \(\approx\) \(1.692770467\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.20 + 1.65i)T \)
good7 \( 1 + (-1.29 + 4.83i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.26 + 4.72i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.47 + 2.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.97 - 1.06i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.451 - 0.782i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.66 + 2.69i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.05 - 4.05i)T - 31iT^{2} \)
37 \( 1 + (4.50 - 1.20i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (6.53 - 1.74i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.10 + 2.94i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.65 - 3.65i)T - 47iT^{2} \)
53 \( 1 - 9.56iT - 53T^{2} \)
59 \( 1 + (8.99 + 2.40i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.81 - 4.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.99 + 7.44i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.55 + 9.55i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.73 + 3.73i)T + 73iT^{2} \)
79 \( 1 - 9.42T + 79T^{2} \)
83 \( 1 + (-10.7 - 10.7i)T + 83iT^{2} \)
89 \( 1 + (-2.43 - 9.07i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.38 + 0.908i)T + (84.0 + 48.5i)T^{2} \)
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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

−8.667842145245319413050989234852, −7.81750510042862965675216733846, −7.35831066960554895777891071084, −6.42983648394233801329160542742, −5.62691797116639731824952078740, −4.76520000453619945129987319535, −3.61964723056827471228999607713, −3.21979033922632254909558632913, −1.56797956548190633746465127602, −0.56936024968224036452861685009, 1.76742093394310229242937915982, 2.16096764376733059097460067156, 3.47626324787892233825290112883, 4.66709772341705221198848060990, 5.31796178038244288009540787177, 5.95022674401104440578050826076, 6.88293314154589071983942022362, 7.82228139758997803870377518304, 8.629463494391356165842288342336, 9.148396458703419679699615700961

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