Properties

Label 2-1815-1.1-c3-0-126
Degree $2$
Conductor $1815$
Sign $1$
Analytic cond. $107.088$
Root an. cond. $10.3483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.46·2-s + 3·3-s + 3.99·4-s + 5·5-s + 10.3·6-s + 31.1·7-s − 13.8·8-s + 9·9-s + 17.3·10-s + 11.9·12-s − 76.2·13-s + 108·14-s + 15·15-s − 80·16-s + 46.7·17-s + 31.1·18-s − 31.1·19-s + 19.9·20-s + 93.5·21-s + 195·23-s − 41.5·24-s + 25·25-s − 264·26-s + 27·27-s + 124.·28-s + 34.6·29-s + 51.9·30-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.577·3-s + 0.499·4-s + 0.447·5-s + 0.707·6-s + 1.68·7-s − 0.612·8-s + 0.333·9-s + 0.547·10-s + 0.288·12-s − 1.62·13-s + 2.06·14-s + 0.258·15-s − 1.25·16-s + 0.667·17-s + 0.408·18-s − 0.376·19-s + 0.223·20-s + 0.971·21-s + 1.76·23-s − 0.353·24-s + 0.200·25-s − 1.99·26-s + 0.192·27-s + 0.841·28-s + 0.221·29-s + 0.316·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(107.088\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.731441860\)
\(L(\frac12)\) \(\approx\) \(6.731441860\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 - 5T \)
11 \( 1 \)
good2 \( 1 - 3.46T + 8T^{2} \)
7 \( 1 - 31.1T + 343T^{2} \)
13 \( 1 + 76.2T + 2.19e3T^{2} \)
17 \( 1 - 46.7T + 4.91e3T^{2} \)
19 \( 1 + 31.1T + 6.85e3T^{2} \)
23 \( 1 - 195T + 1.21e4T^{2} \)
29 \( 1 - 34.6T + 2.43e4T^{2} \)
31 \( 1 - 97T + 2.97e4T^{2} \)
37 \( 1 - 274T + 5.06e4T^{2} \)
41 \( 1 - 384.T + 6.89e4T^{2} \)
43 \( 1 - 415.T + 7.95e4T^{2} \)
47 \( 1 - 3T + 1.03e5T^{2} \)
53 \( 1 - 3T + 1.48e5T^{2} \)
59 \( 1 + 648T + 2.05e5T^{2} \)
61 \( 1 + 458.T + 2.26e5T^{2} \)
67 \( 1 - 70T + 3.00e5T^{2} \)
71 \( 1 + 372T + 3.57e5T^{2} \)
73 \( 1 - 935.T + 3.89e5T^{2} \)
79 \( 1 - 521.T + 4.93e5T^{2} \)
83 \( 1 - 45.0T + 5.71e5T^{2} \)
89 \( 1 - 1.12e3T + 7.04e5T^{2} \)
97 \( 1 - 470T + 9.12e5T^{2} \)
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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

−9.020415492340436091207147850099, −7.898330417445382248980147621505, −7.43230277029857419057003768353, −6.30803203395439759784562414877, −5.28621874690842429323146325444, −4.80757806477824896988595153503, −4.20375464771236087929042708642, −2.85085708624874405759539496436, −2.32741617231247832729762673611, −1.03848987213083492164566398181, 1.03848987213083492164566398181, 2.32741617231247832729762673611, 2.85085708624874405759539496436, 4.20375464771236087929042708642, 4.80757806477824896988595153503, 5.28621874690842429323146325444, 6.30803203395439759784562414877, 7.43230277029857419057003768353, 7.898330417445382248980147621505, 9.020415492340436091207147850099

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