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Properties

Label 2-1305-5.4-c1-0-24
Degree $2$
Conductor $1305$
Sign $0.575 - 0.817i$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
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  − 0.156i·2-s + 1.97·4-s + (−1.28 + 1.82i)5-s − 1.09i·7-s − 0.623i·8-s + (0.286 + 0.201i)10-s + 4.40·11-s + 3.97i·13-s − 0.171·14-s + 3.85·16-s + 6.22i·17-s − 6.97·19-s + (−2.54 + 3.61i)20-s − 0.690i·22-s + 0.780i·23-s + ⋯
L(s)  = 1  − 0.110i·2-s + 0.987·4-s + (−0.575 + 0.817i)5-s − 0.413i·7-s − 0.220i·8-s + (0.0906 + 0.0637i)10-s + 1.32·11-s + 1.10i·13-s − 0.0458·14-s + 0.963·16-s + 1.50i·17-s − 1.60·19-s + (−0.568 + 0.807i)20-s − 0.147i·22-s + 0.162i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.575 - 0.817i$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 0.575 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.927869575\)
\(L(\frac12)\) \(\approx\) \(1.927869575\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 - 1.82i)T \)
29 \( 1 + T \)
good2 \( 1 + 0.156iT - 2T^{2} \)
7 \( 1 + 1.09iT - 7T^{2} \)
11 \( 1 - 4.40T + 11T^{2} \)
13 \( 1 - 3.97iT - 13T^{2} \)
17 \( 1 - 6.22iT - 17T^{2} \)
19 \( 1 + 6.97T + 19T^{2} \)
23 \( 1 - 0.780iT - 23T^{2} \)
31 \( 1 - 6.40T + 31T^{2} \)
37 \( 1 - 1.09iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 0.376iT - 43T^{2} \)
47 \( 1 + 4.75iT - 47T^{2} \)
53 \( 1 - 11.2iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 1.14T + 61T^{2} \)
67 \( 1 - 5.90iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 8.72iT - 73T^{2} \)
79 \( 1 - 5.54T + 79T^{2} \)
83 \( 1 + 6.22iT - 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 17.4iT - 97T^{2} \)
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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

−10.07270060743804332958132930519, −8.855347678350735152666523664304, −8.097995221158992988512362284975, −7.06222094954113082768768569278, −6.58731441746698825582141495976, −6.05014955958560509747000476029, −4.11713908680698439728966498775, −3.90199322259851344905446783997, −2.51511411204060374085232249140, −1.49877639899808921747178737467, 0.832729770966936186513071122928, 2.21276395948240283661252444857, 3.32157852128560787664348327722, 4.40789994873025905437514421584, 5.37322401488110472540262596178, 6.29322061478904870846402049832, 7.03009278888006840627929804395, 7.958412690487321856105129708969, 8.605103198488243247445698829674, 9.412784290887377307105659624769

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