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Properties

Label 2-465-15.2-c1-0-30
Degree $2$
Conductor $465$
Sign $0.762 + 0.647i$
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  + (1.13 − 1.13i)2-s + (−1.70 − 0.328i)3-s − 0.573i·4-s + (1.95 − 1.08i)5-s + (−2.30 + 1.55i)6-s + (1.84 + 1.84i)7-s + (1.61 + 1.61i)8-s + (2.78 + 1.11i)9-s + (0.995 − 3.44i)10-s + 1.80i·11-s + (−0.188 + 0.975i)12-s + (−2.48 + 2.48i)13-s + 4.18·14-s + (−3.68 + 1.19i)15-s + 4.81·16-s + (2.49 − 2.49i)17-s + ⋯
L(s)  = 1  + (0.802 − 0.802i)2-s + (−0.981 − 0.189i)3-s − 0.286i·4-s + (0.875 − 0.483i)5-s + (−0.939 + 0.635i)6-s + (0.697 + 0.697i)7-s + (0.572 + 0.572i)8-s + (0.928 + 0.372i)9-s + (0.314 − 1.08i)10-s + 0.543i·11-s + (−0.0544 + 0.281i)12-s + (−0.688 + 0.688i)13-s + 1.11·14-s + (−0.951 + 0.308i)15-s + 1.20·16-s + (0.605 − 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.762 + 0.647i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.762 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87965 - 0.690337i\)
\(L(\frac12)\) \(\approx\) \(1.87965 - 0.690337i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.328i)T \)
5 \( 1 + (-1.95 + 1.08i)T \)
31 \( 1 + T \)
good2 \( 1 + (-1.13 + 1.13i)T - 2iT^{2} \)
7 \( 1 + (-1.84 - 1.84i)T + 7iT^{2} \)
11 \( 1 - 1.80iT - 11T^{2} \)
13 \( 1 + (2.48 - 2.48i)T - 13iT^{2} \)
17 \( 1 + (-2.49 + 2.49i)T - 17iT^{2} \)
19 \( 1 + 3.52iT - 19T^{2} \)
23 \( 1 + (2.67 + 2.67i)T + 23iT^{2} \)
29 \( 1 - 2.97T + 29T^{2} \)
37 \( 1 + (-3.47 - 3.47i)T + 37iT^{2} \)
41 \( 1 + 5.74iT - 41T^{2} \)
43 \( 1 + (1.97 - 1.97i)T - 43iT^{2} \)
47 \( 1 + (8.92 - 8.92i)T - 47iT^{2} \)
53 \( 1 + (1.83 + 1.83i)T + 53iT^{2} \)
59 \( 1 - 8.71T + 59T^{2} \)
61 \( 1 + 7.80T + 61T^{2} \)
67 \( 1 + (5.57 + 5.57i)T + 67iT^{2} \)
71 \( 1 - 2.87iT - 71T^{2} \)
73 \( 1 + (5.07 - 5.07i)T - 73iT^{2} \)
79 \( 1 - 6.87iT - 79T^{2} \)
83 \( 1 + (11.9 + 11.9i)T + 83iT^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + (5.53 + 5.53i)T + 97iT^{2} \)
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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

−11.31675750107883666258896070344, −10.26502284021790357080093380023, −9.479361128427258138648093207121, −8.221898398920059512309150357258, −7.05795129539872700649285584231, −5.86922979048536247308740677920, −4.89531432877945089842020678186, −4.58522004564283649982684438886, −2.55374103120521554454257784122, −1.61325333338222965068394872540, 1.39726572699076283334931725799, 3.61423878001565160289317827005, 4.77114197391337580454320240699, 5.61053411624395680883089182606, 6.15392452503067374496952533472, 7.16396058198181146841411895699, 7.971321921292091592720529698293, 9.819018150380868527307886344826, 10.27466427056302385228119414985, 10.99432204569808246238857104502

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