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Properties

Label 2-345-23.8-c1-0-4
Degree $2$
Conductor $345$
Sign $-0.964 - 0.263i$
Analytic cond. $2.75483$
Root an. cond. $1.65977$
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  + (−0.290 + 2.01i)2-s + (0.415 + 0.909i)3-s + (−2.06 − 0.606i)4-s + (0.654 + 0.755i)5-s + (−1.95 + 0.574i)6-s + (3.11 + 2.00i)7-s + (0.130 − 0.286i)8-s + (−0.654 + 0.755i)9-s + (−1.71 + 1.10i)10-s + (−0.0961 − 0.668i)11-s + (−0.306 − 2.13i)12-s + (2.80 − 1.80i)13-s + (−4.94 + 5.70i)14-s + (−0.415 + 0.909i)15-s + (−3.08 − 1.98i)16-s + (0.151 − 0.0445i)17-s + ⋯
L(s)  = 1  + (−0.205 + 1.42i)2-s + (0.239 + 0.525i)3-s + (−1.03 − 0.303i)4-s + (0.292 + 0.337i)5-s + (−0.798 + 0.234i)6-s + (1.17 + 0.756i)7-s + (0.0462 − 0.101i)8-s + (−0.218 + 0.251i)9-s + (−0.542 + 0.348i)10-s + (−0.0289 − 0.201i)11-s + (−0.0885 − 0.615i)12-s + (0.779 − 0.500i)13-s + (−1.32 + 1.52i)14-s + (−0.107 + 0.234i)15-s + (−0.771 − 0.495i)16-s + (0.0367 − 0.0108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $-0.964 - 0.263i$
Analytic conductor: \(2.75483\)
Root analytic conductor: \(1.65977\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{345} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 345,\ (\ :1/2),\ -0.964 - 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192259 + 1.43277i\)
\(L(\frac12)\) \(\approx\) \(0.192259 + 1.43277i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.415 - 0.909i)T \)
5 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (4.57 + 1.43i)T \)
good2 \( 1 + (0.290 - 2.01i)T + (-1.91 - 0.563i)T^{2} \)
7 \( 1 + (-3.11 - 2.00i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (0.0961 + 0.668i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (-2.80 + 1.80i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-0.151 + 0.0445i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (0.499 + 0.146i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (6.82 - 2.00i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-3.12 + 6.85i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-3.49 + 4.02i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (-2.98 - 3.44i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (2.97 + 6.50i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 + (0.416 + 0.267i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-10.0 + 6.48i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (3.44 - 7.53i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (2.03 - 14.1i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (1.48 - 10.3i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-3.69 - 1.08i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (5.93 - 3.81i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-1.22 + 1.40i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (2.68 + 5.88i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-3.02 - 3.48i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.67612073488643353533671455846, −11.00285896496602167633140282855, −9.803713938887464030268783798386, −8.729739875598670839142071290180, −8.236092631775312781036154385999, −7.31496034473732910830159170112, −5.90603485269118971445350177065, −5.51041876134942770148283691962, −4.17460815397302623798413530343, −2.36520587408193625121178987720, 1.21281637252009293078407059638, 2.06523306308800141396322446469, 3.66349278526923178176897858757, 4.66404029904859013227761936428, 6.23904192977714579517785820825, 7.54952485506401408277165900487, 8.482527458145639893563875124942, 9.392798729445463234429674991514, 10.38689883387952480985917231095, 11.17083426462733629723312262339

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