Properties

Label 2-495-11.9-c1-0-15
Degree $2$
Conductor $495$
Sign $0.206 + 0.978i$
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  + (0.372 − 1.14i)2-s + (0.441 + 0.320i)4-s + (0.309 + 0.951i)5-s + (−3.49 − 2.53i)7-s + (2.48 − 1.80i)8-s + 1.20·10-s + (2.96 + 1.49i)11-s + (1.91 − 5.87i)13-s + (−4.20 + 3.05i)14-s + (−0.806 − 2.48i)16-s + (−0.248 − 0.764i)17-s + (1.51 − 1.10i)19-s + (−0.168 + 0.519i)20-s + (2.81 − 2.84i)22-s + 8.08·23-s + ⋯
L(s)  = 1  + (0.263 − 0.810i)2-s + (0.220 + 0.160i)4-s + (0.138 + 0.425i)5-s + (−1.31 − 0.958i)7-s + (0.878 − 0.637i)8-s + 0.381·10-s + (0.893 + 0.449i)11-s + (0.529 − 1.63i)13-s + (−1.12 + 0.817i)14-s + (−0.201 − 0.620i)16-s + (−0.0602 − 0.185i)17-s + (0.347 − 0.252i)19-s + (−0.0377 + 0.116i)20-s + (0.600 − 0.605i)22-s + 1.68·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.206 + 0.978i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.206 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39926 - 1.13536i\)
\(L(\frac12)\) \(\approx\) \(1.39926 - 1.13536i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.96 - 1.49i)T \)
good2 \( 1 + (-0.372 + 1.14i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (3.49 + 2.53i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.91 + 5.87i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.248 + 0.764i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.51 + 1.10i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 8.08T + 23T^{2} \)
29 \( 1 + (-0.857 - 0.622i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.499 - 1.53i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.48 - 3.25i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.08 - 2.96i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.87T + 43T^{2} \)
47 \( 1 + (10.5 - 7.63i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.823 - 2.53i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.34 + 3.88i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.65 + 5.08i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.49T + 67T^{2} \)
71 \( 1 + (1.80 + 5.54i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.88 + 2.82i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.31 - 13.2i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.72 - 11.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (-0.416 + 1.28i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.76058782004880733774033730719, −10.09925282440076463401694493701, −9.386687067441628560869264890089, −7.88224643099921857729687029843, −6.92935642617613091991052392294, −6.39302418415157810300000990406, −4.74177142035892673209133460235, −3.32089613879751244545264599684, −3.13451845939131824923883872048, −1.13862353739453269143888065578, 1.73692072742484326813826045931, 3.32885318848866544365520580184, 4.68914954002362728902465427317, 5.84910297076287206584277474660, 6.43163408739860903700802188683, 7.10199428754260813779508471693, 8.669888134807646261250799800209, 9.090879291560768329903541260782, 10.06446771584735450978098700157, 11.40452633177147256346080986418

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