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Properties

Label 2-1815-5.4-c1-0-9
Degree $2$
Conductor $1815$
Sign $0.929 - 0.368i$
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  − 2.10i·2-s + i·3-s − 2.43·4-s + (−0.823 − 2.07i)5-s + 2.10·6-s + 3.31i·7-s + 0.921i·8-s − 9-s + (−4.37 + 1.73i)10-s − 2.43i·12-s + 2.49i·13-s + 6.98·14-s + (2.07 − 0.823i)15-s − 2.93·16-s − 6.81i·17-s + 2.10i·18-s + ⋯
L(s)  = 1  − 1.48i·2-s + 0.577i·3-s − 1.21·4-s + (−0.368 − 0.929i)5-s + 0.859·6-s + 1.25i·7-s + 0.325i·8-s − 0.333·9-s + (−1.38 + 0.548i)10-s − 0.703i·12-s + 0.693i·13-s + 1.86·14-s + (0.536 − 0.212i)15-s − 0.733·16-s − 1.65i·17-s + 0.496i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.929 - 0.368i$
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: \((2,\ 1815,\ (\ :1/2),\ 0.929 - 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8078828078\)
\(L(\frac12)\) \(\approx\) \(0.8078828078\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.823 + 2.07i)T \)
11 \( 1 \)
good2 \( 1 + 2.10iT - 2T^{2} \)
7 \( 1 - 3.31iT - 7T^{2} \)
13 \( 1 - 2.49iT - 13T^{2} \)
17 \( 1 + 6.81iT - 17T^{2} \)
19 \( 1 + 1.14T + 19T^{2} \)
23 \( 1 - 5.89iT - 23T^{2} \)
29 \( 1 + 1.12T + 29T^{2} \)
31 \( 1 + 1.09T + 31T^{2} \)
37 \( 1 - 11.5iT - 37T^{2} \)
41 \( 1 - 0.932T + 41T^{2} \)
43 \( 1 + 0.552iT - 43T^{2} \)
47 \( 1 - 2.13iT - 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 + 8.39T + 59T^{2} \)
61 \( 1 - 8.21T + 61T^{2} \)
67 \( 1 - 4.15iT - 67T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 - 5.73iT - 73T^{2} \)
79 \( 1 + 3.86T + 79T^{2} \)
83 \( 1 - 6.08iT - 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 - 2.86iT - 97T^{2} \)
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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

−9.408841926252484728634422462588, −8.999446446807938950881155017856, −8.178869326982245058837169853927, −6.98585353462304639864008366233, −5.67309633084753131264951700534, −4.93504745296704267190425710392, −4.24795119890212092558123881477, −3.23083595165918815081728399178, −2.41609296281738793482431895368, −1.27480162265684293033807776441, 0.31260005055185178316632862153, 2.15737427597957223782098877631, 3.58580842020591784726624017427, 4.31350258918768780637218809725, 5.55502459749116987807697241395, 6.35471005383231305131812032926, 6.83269456511005756688426164932, 7.56256547999872479771842501965, 8.032024283557370799031267357187, 8.718122090295921150754106510108

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