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Properties

Label 2-465-15.8-c1-0-51
Degree $2$
Conductor $465$
Sign $0.977 + 0.209i$
Analytic cond. $3.71304$
Root an. cond. $1.92692$
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  + (0.814 + 0.814i)2-s + (1.69 + 0.360i)3-s − 0.673i·4-s + (−1.17 − 1.90i)5-s + (1.08 + 1.67i)6-s + (−0.307 + 0.307i)7-s + (2.17 − 2.17i)8-s + (2.74 + 1.22i)9-s + (0.595 − 2.50i)10-s − 4.98i·11-s + (0.242 − 1.14i)12-s + (−2.07 − 2.07i)13-s − 0.500·14-s + (−1.30 − 3.64i)15-s + 2.20·16-s + (4.37 + 4.37i)17-s + ⋯
L(s)  = 1  + (0.575 + 0.575i)2-s + (0.978 + 0.207i)3-s − 0.336i·4-s + (−0.524 − 0.851i)5-s + (0.443 + 0.683i)6-s + (−0.116 + 0.116i)7-s + (0.769 − 0.769i)8-s + (0.913 + 0.406i)9-s + (0.188 − 0.792i)10-s − 1.50i·11-s + (0.0699 − 0.329i)12-s + (−0.576 − 0.576i)13-s − 0.133·14-s + (−0.336 − 0.941i)15-s + 0.550·16-s + (1.06 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.977 + 0.209i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.977 + 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40828 - 0.254834i\)
\(L(\frac12)\) \(\approx\) \(2.40828 - 0.254834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 - 0.360i)T \)
5 \( 1 + (1.17 + 1.90i)T \)
31 \( 1 + T \)
good2 \( 1 + (-0.814 - 0.814i)T + 2iT^{2} \)
7 \( 1 + (0.307 - 0.307i)T - 7iT^{2} \)
11 \( 1 + 4.98iT - 11T^{2} \)
13 \( 1 + (2.07 + 2.07i)T + 13iT^{2} \)
17 \( 1 + (-4.37 - 4.37i)T + 17iT^{2} \)
19 \( 1 - 6.03iT - 19T^{2} \)
23 \( 1 + (2.88 - 2.88i)T - 23iT^{2} \)
29 \( 1 - 1.02T + 29T^{2} \)
37 \( 1 + (1.26 - 1.26i)T - 37iT^{2} \)
41 \( 1 + 7.87iT - 41T^{2} \)
43 \( 1 + (-6.46 - 6.46i)T + 43iT^{2} \)
47 \( 1 + (-1.72 - 1.72i)T + 47iT^{2} \)
53 \( 1 + (1.46 - 1.46i)T - 53iT^{2} \)
59 \( 1 + 8.08T + 59T^{2} \)
61 \( 1 + 0.446T + 61T^{2} \)
67 \( 1 + (-2.60 + 2.60i)T - 67iT^{2} \)
71 \( 1 + 4.96iT - 71T^{2} \)
73 \( 1 + (-7.21 - 7.21i)T + 73iT^{2} \)
79 \( 1 + 13.0iT - 79T^{2} \)
83 \( 1 + (0.162 - 0.162i)T - 83iT^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + (1.24 - 1.24i)T - 97iT^{2} \)
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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

−10.81287236978766463378742923437, −10.02448644804030964097973412720, −9.109939679776621444878021864632, −8.021858364490702272473710850600, −7.70623131717331062998032592040, −6.04146562020012660429411593501, −5.39668060522277578859276671600, −4.12372896571346432960758126249, −3.37875095661580168570440266294, −1.34833454346943358640098051858, 2.21614552269871421172253017623, 2.91562594904080861621923263850, 4.04231111654573147457873470177, 4.80659626292308237221083994437, 6.99728563356244076214138848499, 7.24270285968844862446466765553, 8.186421666784223921537573850100, 9.444043211624139284888395299130, 10.15223267552143173957622583856, 11.28585193416883029548526095978

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