Properties

Label 2-345-23.4-c1-0-11
Degree $2$
Conductor $345$
Sign $0.977 + 0.212i$
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  + (0.582 + 1.27i)2-s + (−0.959 + 0.281i)3-s + (0.0227 − 0.0262i)4-s + (−0.841 − 0.540i)5-s + (−0.918 − 1.05i)6-s + (−0.746 − 5.19i)7-s + (2.73 + 0.803i)8-s + (0.841 − 0.540i)9-s + (0.199 − 1.38i)10-s + (0.563 − 1.23i)11-s + (−0.0144 + 0.0315i)12-s + (0.515 − 3.58i)13-s + (6.18 − 3.97i)14-s + (0.959 + 0.281i)15-s + (0.559 + 3.88i)16-s + (3.18 + 3.68i)17-s + ⋯
L(s)  = 1  + (0.411 + 0.901i)2-s + (−0.553 + 0.162i)3-s + (0.0113 − 0.0131i)4-s + (−0.376 − 0.241i)5-s + (−0.374 − 0.432i)6-s + (−0.282 − 1.96i)7-s + (0.967 + 0.284i)8-s + (0.280 − 0.180i)9-s + (0.0630 − 0.438i)10-s + (0.169 − 0.371i)11-s + (−0.00416 + 0.00911i)12-s + (0.142 − 0.993i)13-s + (1.65 − 1.06i)14-s + (0.247 + 0.0727i)15-s + (0.139 + 0.972i)16-s + (0.773 + 0.892i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $0.977 + 0.212i$
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: \((2,\ 345,\ (\ :1/2),\ 0.977 + 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37368 - 0.147505i\)
\(L(\frac12)\) \(\approx\) \(1.37368 - 0.147505i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.40 + 4.14i)T \)
good2 \( 1 + (-0.582 - 1.27i)T + (-1.30 + 1.51i)T^{2} \)
7 \( 1 + (0.746 + 5.19i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-0.563 + 1.23i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.515 + 3.58i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-3.18 - 3.68i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (2.65 - 3.05i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (2.04 + 2.35i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-2.24 - 0.657i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-3.55 + 2.28i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-9.25 - 5.94i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (0.153 - 0.0451i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 0.618T + 47T^{2} \)
53 \( 1 + (0.859 + 5.97i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (0.998 - 6.94i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-7.19 - 2.11i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (6.44 + 14.1i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-5.65 - 12.3i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (9.28 - 10.7i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-1.04 + 7.24i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-4.36 + 2.80i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (7.61 - 2.23i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-12.1 - 7.83i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.25726853828083140280109795492, −10.49381850500250328816504936710, −10.05434736240304181442989877264, −8.119370772674188134287453871511, −7.61224280206129154896465231683, −6.50378042055427025365530625798, −5.81351359698684685368855484713, −4.47393358871790283966976437324, −3.77737334447653784879887666715, −0.975995086705516184989610406529, 1.98761833400954276118448396979, 2.99614937708939356498925875755, 4.39228587842799147030072978216, 5.52662394159049488937578473291, 6.63074825276190385358070653227, 7.66807063591936558103693662515, 9.004673906293635724594954224529, 9.796749040051719337437664763259, 11.13668805958798555533313970478, 11.67113275557379543324828740817

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