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Properties

Label 2-456-152.75-c1-0-20
Degree $2$
Conductor $456$
Sign $0.709 + 0.704i$
Analytic cond. $3.64117$
Root an. cond. $1.90818$
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  + (0.134 − 1.40i)2-s + i·3-s + (−1.96 − 0.378i)4-s + 2.61i·5-s + (1.40 + 0.134i)6-s − 4.81i·7-s + (−0.797 + 2.71i)8-s − 9-s + (3.68 + 0.351i)10-s + 3.49·11-s + (0.378 − 1.96i)12-s + 1.50·13-s + (−6.77 − 0.647i)14-s − 2.61·15-s + (3.71 + 1.48i)16-s + 5.31·17-s + ⋯
L(s)  = 1  + (0.0951 − 0.995i)2-s + 0.577i·3-s + (−0.981 − 0.189i)4-s + 1.16i·5-s + (0.574 + 0.0549i)6-s − 1.81i·7-s + (−0.281 + 0.959i)8-s − 0.333·9-s + (1.16 + 0.111i)10-s + 1.05·11-s + (0.109 − 0.566i)12-s + 0.416·13-s + (−1.81 − 0.173i)14-s − 0.675·15-s + (0.928 + 0.371i)16-s + 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.709 + 0.704i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32358 - 0.545076i\)
\(L(\frac12)\) \(\approx\) \(1.32358 - 0.545076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.134 + 1.40i)T \)
3 \( 1 - iT \)
19 \( 1 + (-3.83 - 2.07i)T \)
good5 \( 1 - 2.61iT - 5T^{2} \)
7 \( 1 + 4.81iT - 7T^{2} \)
11 \( 1 - 3.49T + 11T^{2} \)
13 \( 1 - 1.50T + 13T^{2} \)
17 \( 1 - 5.31T + 17T^{2} \)
23 \( 1 + 2.26iT - 23T^{2} \)
29 \( 1 - 5.12T + 29T^{2} \)
31 \( 1 - 1.72T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 3.90iT - 41T^{2} \)
43 \( 1 - 0.342T + 43T^{2} \)
47 \( 1 + 0.679iT - 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 + 9.15iT - 59T^{2} \)
61 \( 1 + 2.65iT - 61T^{2} \)
67 \( 1 - 7.37iT - 67T^{2} \)
71 \( 1 - 9.74T + 71T^{2} \)
73 \( 1 + 7.16T + 73T^{2} \)
79 \( 1 + 5.21T + 79T^{2} \)
83 \( 1 + 4.80T + 83T^{2} \)
89 \( 1 - 14.0iT - 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−10.87309431293632647891922960504, −10.12602482379915397710133248172, −9.792749339649904830700660788703, −8.373659630826935362484980659915, −7.28908301050922923944673769927, −6.27640665268440719584657270822, −4.78672538953901945922570521346, −3.67068123798269844778810344696, −3.27088832383080914792372486566, −1.20116166955081659435053293113, 1.29622503760274973959272965329, 3.29035788875076704153171080116, 4.90429171943358455487772981149, 5.58287312286423703222766695093, 6.35731270374105741929203610091, 7.57507235574885134090873821965, 8.673924525071643810333068223421, 8.861298687566825773584811094129, 9.774606805648704494269310823860, 11.76387656045233807494934551083

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