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Properties

Label 2-1815-5.4-c1-0-3
Degree $2$
Conductor $1815$
Sign $0.894 + 0.447i$
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  + 2.23i·2-s + i·3-s − 3.00·4-s + (−1 + 2i)5-s − 2.23·6-s − 2.23i·8-s − 9-s + (−4.47 − 2.23i)10-s − 3.00i·12-s + 4.47i·13-s + (−2 − i)15-s − 0.999·16-s − 4.47i·17-s − 2.23i·18-s + (3.00 − 6.00i)20-s + ⋯
L(s)  = 1  + 1.58i·2-s + 0.577i·3-s − 1.50·4-s + (−0.447 + 0.894i)5-s − 0.912·6-s − 0.790i·8-s − 0.333·9-s + (−1.41 − 0.707i)10-s − 0.866i·12-s + 1.24i·13-s + (−0.516 − 0.258i)15-s − 0.249·16-s − 1.08i·17-s − 0.527i·18-s + (0.670 − 1.34i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.894 + 0.447i$
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: \((2,\ 1815,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3941379753\)
\(L(\frac12)\) \(\approx\) \(0.3941379753\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
11 \( 1 \)
good2 \( 1 - 2.23iT - 2T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 - 4.47iT - 13T^{2} \)
17 \( 1 + 4.47iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 8.94T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 8.94T + 41T^{2} \)
43 \( 1 + 8.94iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 13.4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.673110340659262307730827396729, −9.044236019170433713133632072097, −8.278226189809698174789571321051, −7.40426823121631483002607118915, −6.87080100082620743430596886949, −6.22975913797124368422794781131, −5.23819242570815352755350787687, −4.47653462927509338405714176869, −3.63651926977594442312264451213, −2.37601141727759994178201477679, 0.15212646579063749017579757646, 1.27889281267816482585358764778, 2.13216109295891719054374850049, 3.41478905759745497837594884478, 3.92203046049116926398319652469, 5.13313972688726059222824635259, 5.80908696434550465916362382156, 7.18566371317146774162636057407, 8.012655547793926765874404545616, 8.666220609108065011629930803816

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