Properties

Label 2-1815-5.4-c1-0-92
Degree $2$
Conductor $1815$
Sign $-0.659 + 0.751i$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
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  + 0.298i·2-s i·3-s + 1.91·4-s + (−1.68 − 1.47i)5-s + 0.298·6-s − 2.32i·7-s + 1.16i·8-s − 9-s + (0.439 − 0.501i)10-s − 1.91i·12-s − 4.59i·13-s + 0.691·14-s + (−1.47 + 1.68i)15-s + 3.47·16-s − 5.14i·17-s − 0.298i·18-s + ⋯
L(s)  = 1  + 0.210i·2-s − 0.577i·3-s + 0.955·4-s + (−0.751 − 0.659i)5-s + 0.121·6-s − 0.877i·7-s + 0.412i·8-s − 0.333·9-s + (0.138 − 0.158i)10-s − 0.551i·12-s − 1.27i·13-s + 0.184·14-s + (−0.380 + 0.434i)15-s + 0.868·16-s − 1.24i·17-s − 0.0702i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.659 + 0.751i$
Analytic conductor: \(14.4928\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ -0.659 + 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.570928153\)
\(L(\frac12)\) \(\approx\) \(1.570928153\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (1.68 + 1.47i)T \)
11 \( 1 \)
good2 \( 1 - 0.298iT - 2T^{2} \)
7 \( 1 + 2.32iT - 7T^{2} \)
13 \( 1 + 4.59iT - 13T^{2} \)
17 \( 1 + 5.14iT - 17T^{2} \)
19 \( 1 - 5.69T + 19T^{2} \)
23 \( 1 - 6.02iT - 23T^{2} \)
29 \( 1 + 6.09T + 29T^{2} \)
31 \( 1 + 8.43T + 31T^{2} \)
37 \( 1 - 3.17iT - 37T^{2} \)
41 \( 1 + 0.468T + 41T^{2} \)
43 \( 1 + 8.08iT - 43T^{2} \)
47 \( 1 + 9.18iT - 47T^{2} \)
53 \( 1 + 4.25iT - 53T^{2} \)
59 \( 1 + 6.65T + 59T^{2} \)
61 \( 1 - 1.82T + 61T^{2} \)
67 \( 1 - 8.86iT - 67T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 + 9.08iT - 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 8.34iT - 83T^{2} \)
89 \( 1 - 3.73T + 89T^{2} \)
97 \( 1 + 5.58iT - 97T^{2} \)
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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

−8.809215811826255132092345484369, −7.75361195626458007182392663747, −7.43341261311097905734800282797, −7.03031554462724560740619456853, −5.52654800139796586425350447531, −5.30280072782043601484088011402, −3.71405532950931764325114885597, −3.08860150459106067456452959241, −1.62450361998969511796803246872, −0.55902054686764270028575643218, 1.76999676178350108995572768914, 2.75277747474838020453907756656, 3.58894332248835433772110975195, 4.40725084876517213589062029775, 5.70806281006975197585005789752, 6.31707053163034861419528883852, 7.21352150444939121530026778068, 7.88501669702713384279626133261, 8.890528495451628115192272406876, 9.548005112722549468861045632780

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