Properties

Label 2-1815-1.1-c1-0-2
Degree $2$
Conductor $1815$
Sign $1$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.23·2-s − 3-s − 0.477·4-s + 5-s + 1.23·6-s − 3.79·7-s + 3.05·8-s + 9-s − 1.23·10-s + 0.477·12-s − 2.46·13-s + 4.68·14-s − 15-s − 2.81·16-s + 6.52·17-s − 1.23·18-s − 8.25·19-s − 0.477·20-s + 3.79·21-s − 2.34·23-s − 3.05·24-s + 25-s + 3.04·26-s − 27-s + 1.81·28-s − 2.46·29-s + 1.23·30-s + ⋯
L(s)  = 1  − 0.872·2-s − 0.577·3-s − 0.238·4-s + 0.447·5-s + 0.503·6-s − 1.43·7-s + 1.08·8-s + 0.333·9-s − 0.390·10-s + 0.137·12-s − 0.684·13-s + 1.25·14-s − 0.258·15-s − 0.704·16-s + 1.58·17-s − 0.290·18-s − 1.89·19-s − 0.106·20-s + 0.827·21-s − 0.487·23-s − 0.623·24-s + 0.200·25-s + 0.597·26-s − 0.192·27-s + 0.342·28-s − 0.458·29-s + 0.225·30-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.4596960349\)
\(L(\frac12)\) \(\approx\) \(0.4596960349\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + 1.23T + 2T^{2} \)
7 \( 1 + 3.79T + 7T^{2} \)
13 \( 1 + 2.46T + 13T^{2} \)
17 \( 1 - 6.52T + 17T^{2} \)
19 \( 1 + 8.25T + 19T^{2} \)
23 \( 1 + 2.34T + 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 - 5.27T + 31T^{2} \)
37 \( 1 - 3.34T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + 9.56T + 47T^{2} \)
53 \( 1 + 6.61T + 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 6.66T + 61T^{2} \)
67 \( 1 - 3.34T + 67T^{2} \)
71 \( 1 - 7.22T + 71T^{2} \)
73 \( 1 + 9.72T + 73T^{2} \)
79 \( 1 - 9.65T + 79T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−9.560100425017046171604412744327, −8.543975652148080775790915123991, −7.83813450969496069213487625356, −6.81270755357276208768078204530, −6.25546732118144264001826545816, −5.31643890216811223892405705180, −4.36371412397755383991477360608, −3.30495458457395109417196497002, −1.96760625177718074842108364427, −0.53106185488785757090461552161, 0.53106185488785757090461552161, 1.96760625177718074842108364427, 3.30495458457395109417196497002, 4.36371412397755383991477360608, 5.31643890216811223892405705180, 6.25546732118144264001826545816, 6.81270755357276208768078204530, 7.83813450969496069213487625356, 8.543975652148080775790915123991, 9.560100425017046171604412744327

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