Properties

Label 2-345-23.4-c1-0-0
Degree $2$
Conductor $345$
Sign $-0.466 + 0.884i$
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  + (1.16 + 2.55i)2-s + (−0.959 + 0.281i)3-s + (−3.84 + 4.43i)4-s + (−0.841 − 0.540i)5-s + (−1.83 − 2.11i)6-s + (0.000825 + 0.00574i)7-s + (−10.4 − 3.05i)8-s + (0.841 − 0.540i)9-s + (0.399 − 2.77i)10-s + (−0.519 + 1.13i)11-s + (2.43 − 5.33i)12-s + (−0.321 + 2.23i)13-s + (−0.0136 + 0.00879i)14-s + (0.959 + 0.281i)15-s + (−2.66 − 18.5i)16-s + (−0.102 − 0.117i)17-s + ⋯
L(s)  = 1  + (0.823 + 1.80i)2-s + (−0.553 + 0.162i)3-s + (−1.92 + 2.21i)4-s + (−0.376 − 0.241i)5-s + (−0.749 − 0.865i)6-s + (0.000312 + 0.00217i)7-s + (−3.68 − 1.08i)8-s + (0.280 − 0.180i)9-s + (0.126 − 0.878i)10-s + (−0.156 + 0.343i)11-s + (0.703 − 1.54i)12-s + (−0.0892 + 0.621i)13-s + (−0.00365 + 0.00235i)14-s + (0.247 + 0.0727i)15-s + (−0.665 − 4.62i)16-s + (−0.0247 − 0.0285i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $-0.466 + 0.884i$
Analytic conductor: \(2.75483\)
Root analytic conductor: \(1.65977\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{345} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 345,\ (\ :1/2),\ -0.466 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.494519 - 0.819975i\)
\(L(\frac12)\) \(\approx\) \(0.494519 - 0.819975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-4.64 + 1.20i)T \)
good2 \( 1 + (-1.16 - 2.55i)T + (-1.30 + 1.51i)T^{2} \)
7 \( 1 + (-0.000825 - 0.00574i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (0.519 - 1.13i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.321 - 2.23i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (0.102 + 0.117i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (4.22 - 4.87i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-5.23 - 6.04i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (0.803 + 0.235i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (9.45 - 6.07i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-4.45 - 2.86i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (8.08 - 2.37i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 6.55T + 47T^{2} \)
53 \( 1 + (0.691 + 4.80i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.180 + 1.25i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-5.69 - 1.67i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-4.31 - 9.45i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-2.35 - 5.14i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-7.19 + 8.29i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (0.843 - 5.86i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-10.9 + 7.01i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-3.11 + 0.914i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (14.2 + 9.13i)T + (40.2 + 88.2i)T^{2} \)
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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

−12.46884928315987269401845957823, −11.63740718057322180631385945718, −10.09433538664817167530231932244, −8.849922928812999348041191926601, −8.185805722901692936979735458955, −7.00643907781412836263254556186, −6.46793138093338835628693498899, −5.23670341989188166433270413720, −4.59426594005867565531590620828, −3.51453942278467339233211252135, 0.55144425713639936350805514329, 2.32989694715983102572898522575, 3.48180534225465172376169326835, 4.63601881461120400511051988212, 5.48935928451719436341446138550, 6.66032543001850007738005791331, 8.423902065586295472438747173230, 9.454306120030959510430223769047, 10.62886047937729080211835595745, 10.87408518396025709158985337146

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