Properties

Label 2-495-11.5-c1-0-16
Degree $2$
Conductor $495$
Sign $-0.356 + 0.934i$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
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  + (−0.359 − 1.10i)2-s + (0.525 − 0.381i)4-s + (0.309 − 0.951i)5-s + (3.46 − 2.51i)7-s + (−2.49 − 1.80i)8-s − 1.16·10-s + (3.15 − 1.00i)11-s + (1.59 + 4.91i)13-s + (−4.03 − 2.92i)14-s + (−0.704 + 2.16i)16-s + (−1.54 + 4.75i)17-s + (−4.53 − 3.29i)19-s + (−0.200 − 0.617i)20-s + (−2.25 − 3.12i)22-s + 0.219·23-s + ⋯
L(s)  = 1  + (−0.253 − 0.781i)2-s + (0.262 − 0.190i)4-s + (0.138 − 0.425i)5-s + (1.31 − 0.952i)7-s + (−0.880 − 0.639i)8-s − 0.367·10-s + (0.952 − 0.304i)11-s + (0.442 + 1.36i)13-s + (−1.07 − 0.782i)14-s + (−0.176 + 0.541i)16-s + (−0.374 + 1.15i)17-s + (−1.03 − 0.755i)19-s + (−0.0448 − 0.138i)20-s + (−0.479 − 0.667i)22-s + 0.0458·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.356 + 0.934i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.356 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.910996 - 1.32285i\)
\(L(\frac12)\) \(\approx\) \(0.910996 - 1.32285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-3.15 + 1.00i)T \)
good2 \( 1 + (0.359 + 1.10i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (-3.46 + 2.51i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.59 - 4.91i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.54 - 4.75i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.53 + 3.29i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.219T + 23T^{2} \)
29 \( 1 + (-5.19 + 3.77i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.874 + 2.69i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.17 - 2.30i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.74 - 3.44i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.90T + 43T^{2} \)
47 \( 1 + (0.192 + 0.139i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.783 + 2.41i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.36 - 4.62i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.50 - 4.62i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + (-3.08 + 9.49i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (11.7 - 8.55i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.47 - 7.61i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.87 - 11.9i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 2.56T + 89T^{2} \)
97 \( 1 + (-0.621 - 1.91i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.91342934192471589123163546198, −9.959970540517548038463099651359, −8.923603465655913562624512770419, −8.293918239690006259286770792737, −6.85229673340528964196116655577, −6.21077935707542024525064951398, −4.57293984215362689039624955035, −3.90945465482265295478297833432, −2.02118384854476781012478782730, −1.20002819559178525208723077006, 1.95867362279268281919616499328, 3.18727888059829190936647715776, 4.86852968654595641876567751191, 5.80212327561604013059904591487, 6.66847160948610476403318172424, 7.64855942917577827550974541890, 8.454474283748839425855556720946, 9.018800292033856174772118946454, 10.42068487302077203449640616264, 11.28011248903383336603920915066

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