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Properties

Label 2-1445-5.4-c1-0-37
Degree $2$
Conductor $1445$
Sign $-0.959 - 0.282i$
Analytic cond. $11.5383$
Root an. cond. $3.39681$
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  + 1.74i·2-s + 0.598i·3-s − 1.03·4-s + (2.14 + 0.632i)5-s − 1.04·6-s + 1.03i·7-s + 1.68i·8-s + 2.64·9-s + (−1.10 + 3.73i)10-s + 0.0764·11-s − 0.620i·12-s − 2.86i·13-s − 1.80·14-s + (−0.378 + 1.28i)15-s − 4.99·16-s + ⋯
L(s)  = 1  + 1.23i·2-s + 0.345i·3-s − 0.517·4-s + (0.959 + 0.282i)5-s − 0.425·6-s + 0.392i·7-s + 0.594i·8-s + 0.880·9-s + (−0.348 + 1.18i)10-s + 0.0230·11-s − 0.178i·12-s − 0.793i·13-s − 0.483·14-s + (−0.0977 + 0.331i)15-s − 1.24·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1445\)    =    \(5 \cdot 17^{2}\)
Sign: $-0.959 - 0.282i$
Analytic conductor: \(11.5383\)
Root analytic conductor: \(3.39681\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1445} (579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1445,\ (\ :1/2),\ -0.959 - 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.161651553\)
\(L(\frac12)\) \(\approx\) \(2.161651553\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.14 - 0.632i)T \)
17 \( 1 \)
good2 \( 1 - 1.74iT - 2T^{2} \)
3 \( 1 - 0.598iT - 3T^{2} \)
7 \( 1 - 1.03iT - 7T^{2} \)
11 \( 1 - 0.0764T + 11T^{2} \)
13 \( 1 + 2.86iT - 13T^{2} \)
19 \( 1 + 4.19T + 19T^{2} \)
23 \( 1 - 5.95iT - 23T^{2} \)
29 \( 1 - 0.0884T + 29T^{2} \)
31 \( 1 + 2.42T + 31T^{2} \)
37 \( 1 - 9.51iT - 37T^{2} \)
41 \( 1 - 8.37T + 41T^{2} \)
43 \( 1 + 0.866iT - 43T^{2} \)
47 \( 1 + 8.33iT - 47T^{2} \)
53 \( 1 - 4.09iT - 53T^{2} \)
59 \( 1 - 9.16T + 59T^{2} \)
61 \( 1 + 0.288T + 61T^{2} \)
67 \( 1 + 12.8iT - 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 - 6.54iT - 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 - 9.70iT - 83T^{2} \)
89 \( 1 + 0.106T + 89T^{2} \)
97 \( 1 - 8.56iT - 97T^{2} \)
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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

−9.751164325156247212509671406419, −9.001605128357064571007602904290, −8.187081621250340645041570662625, −7.28027903154560661871897915083, −6.66512709826185429290353477784, −5.75846264266799452073220611339, −5.30886435190311632965612087142, −4.23641891354721196359694259328, −2.86085253149387742027925611746, −1.72037164832044642728975470468, 0.885187831939022113109303549363, 1.90324547641347786813789841211, 2.54108822924426999796642964313, 4.04785096208631432341209251359, 4.51050419044368862938148265716, 5.95507616660801762880595150671, 6.71819494306668858996281064124, 7.41317075287230328800974591808, 8.775903652582719636986995166318, 9.321903736721722834415715124292

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