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Properties

Label 2-345-115.4-c1-0-11
Degree $2$
Conductor $345$
Sign $0.671 - 0.740i$
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  + (1.19 − 0.544i)2-s + (0.281 + 0.959i)3-s + (−0.185 + 0.213i)4-s + (1.80 + 1.32i)5-s + (0.858 + 0.990i)6-s + (0.0307 − 0.00441i)7-s + (−0.842 + 2.87i)8-s + (−0.841 + 0.540i)9-s + (2.86 + 0.593i)10-s + (−0.773 + 1.69i)11-s + (−0.257 − 0.117i)12-s + (−3.64 − 0.523i)13-s + (0.0341 − 0.0219i)14-s + (−0.760 + 2.10i)15-s + (0.477 + 3.32i)16-s + (3.69 − 3.20i)17-s + ⋯
L(s)  = 1  + (0.842 − 0.384i)2-s + (0.162 + 0.553i)3-s + (−0.0925 + 0.106i)4-s + (0.806 + 0.591i)5-s + (0.350 + 0.404i)6-s + (0.0116 − 0.00166i)7-s + (−0.297 + 1.01i)8-s + (−0.280 + 0.180i)9-s + (0.907 + 0.187i)10-s + (−0.233 + 0.510i)11-s + (−0.0742 − 0.0339i)12-s + (−1.00 − 0.145i)13-s + (0.00913 − 0.00587i)14-s + (−0.196 + 0.542i)15-s + (0.119 + 0.830i)16-s + (0.897 − 0.777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(2.75483\)
Root analytic conductor: \(1.65977\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{345} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 345,\ (\ :1/2),\ 0.671 - 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93776 + 0.858570i\)
\(L(\frac12)\) \(\approx\) \(1.93776 + 0.858570i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.281 - 0.959i)T \)
5 \( 1 + (-1.80 - 1.32i)T \)
23 \( 1 + (0.304 + 4.78i)T \)
good2 \( 1 + (-1.19 + 0.544i)T + (1.30 - 1.51i)T^{2} \)
7 \( 1 + (-0.0307 + 0.00441i)T + (6.71 - 1.97i)T^{2} \)
11 \( 1 + (0.773 - 1.69i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (3.64 + 0.523i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (-3.69 + 3.20i)T + (2.41 - 16.8i)T^{2} \)
19 \( 1 + (-4.16 + 4.80i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-3.14 - 3.63i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-8.92 - 2.62i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (3.98 + 6.19i)T + (-15.3 + 33.6i)T^{2} \)
41 \( 1 + (2.14 + 1.38i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-1.96 - 6.69i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + 8.76iT - 47T^{2} \)
53 \( 1 + (0.438 - 0.0630i)T + (50.8 - 14.9i)T^{2} \)
59 \( 1 + (-1.82 + 12.7i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (6.68 + 1.96i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (7.01 - 3.20i)T + (43.8 - 50.6i)T^{2} \)
71 \( 1 + (3.37 + 7.39i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-1.33 - 1.15i)T + (10.3 + 72.2i)T^{2} \)
79 \( 1 + (0.400 - 2.78i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-4.73 - 7.37i)T + (-34.4 + 75.4i)T^{2} \)
89 \( 1 + (-5.24 + 1.54i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (9.44 - 14.6i)T + (-40.2 - 88.2i)T^{2} \)
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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

−11.80131850105933000036961036767, −10.71107413188940910283065039371, −9.893181403774669247411181102577, −9.118363434947174808865767909970, −7.83057039427043814205894931155, −6.70463941777447956648408166163, −5.21639362071816226848672142285, −4.80911517952746346480065474182, −3.19001725578877845255721743288, −2.51614555873468997777845721251, 1.32605941631828573293932890654, 3.12034946027976636913086976584, 4.58832553156583263059574267918, 5.63944688666111436070132047663, 6.14652083280834080128161144812, 7.47449755292804583930143386912, 8.457942734968651050585563316248, 9.737835078207039469110895768422, 10.10813658211090461788377740588, 11.95914201831626984936377374864

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