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Properties

Label 2-495-11.3-c1-0-1
Degree $2$
Conductor $495$
Sign $-0.955 - 0.296i$
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  + (−1.58 + 1.14i)2-s + (0.564 − 1.73i)4-s + (0.809 + 0.587i)5-s + (0.478 − 1.47i)7-s + (−0.104 − 0.321i)8-s − 1.95·10-s + (−3.11 + 1.14i)11-s + (−3.36 + 2.44i)13-s + (0.935 + 2.87i)14-s + (3.49 + 2.53i)16-s + (0.599 + 0.435i)17-s + (2.31 + 7.11i)19-s + (1.47 − 1.07i)20-s + (3.61 − 5.38i)22-s + 1.15·23-s + ⋯
L(s)  = 1  + (−1.11 + 0.813i)2-s + (0.282 − 0.868i)4-s + (0.361 + 0.262i)5-s + (0.180 − 0.556i)7-s + (−0.0369 − 0.113i)8-s − 0.618·10-s + (−0.938 + 0.343i)11-s + (−0.932 + 0.677i)13-s + (0.249 + 0.769i)14-s + (0.872 + 0.634i)16-s + (0.145 + 0.105i)17-s + (0.530 + 1.63i)19-s + (0.330 − 0.240i)20-s + (0.771 − 1.14i)22-s + 0.241·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.955 - 0.296i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.955 - 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0754076 + 0.497327i\)
\(L(\frac12)\) \(\approx\) \(0.0754076 + 0.497327i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (3.11 - 1.14i)T \)
good2 \( 1 + (1.58 - 1.14i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-0.478 + 1.47i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.36 - 2.44i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.599 - 0.435i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.31 - 7.11i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.15T + 23T^{2} \)
29 \( 1 + (2.12 - 6.54i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.48 - 3.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.21 + 3.72i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.0106 + 0.0328i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + (-2.17 - 6.69i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.990 - 0.719i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.33 + 4.09i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.30 + 3.85i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.55T + 67T^{2} \)
71 \( 1 + (-4.76 - 3.45i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.41 + 4.35i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.99 - 3.63i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (12.2 + 8.90i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + (-0.116 + 0.0848i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.88925446812525828476346067438, −10.17091421062698869258533324846, −9.576568002714753020946870437125, −8.603230143395053656031700439673, −7.50606122675279759717004824169, −7.26219305304914089556241155274, −6.05471712164645984286842100299, −4.98891233777325375083895849644, −3.46021942709840704355527681993, −1.68257574209851841072439221163, 0.42523790219862196662032075800, 2.17904377080936452644385367773, 2.96877370610520550143210196555, 4.98992531234311364003128831889, 5.64777932961080162563063099967, 7.28239550669281241982073496953, 8.134502475905456087454573070859, 8.956906503117618111731961443070, 9.711974881382139734793515192901, 10.37058353620259378193084832806

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