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Properties

Label 2-1815-5.4-c1-0-60
Degree $2$
Conductor $1815$
Sign $-0.405 + 0.914i$
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  − 2.42i·2-s + i·3-s − 3.90·4-s + (2.04 + 0.906i)5-s + 2.42·6-s + 1.34i·7-s + 4.61i·8-s − 9-s + (2.20 − 4.96i)10-s − 3.90i·12-s − 4.12i·13-s + 3.25·14-s + (−0.906 + 2.04i)15-s + 3.41·16-s − 5.87i·17-s + 2.42i·18-s + ⋯
L(s)  = 1  − 1.71i·2-s + 0.577i·3-s − 1.95·4-s + (0.914 + 0.405i)5-s + 0.991·6-s + 0.507i·7-s + 1.63i·8-s − 0.333·9-s + (0.696 − 1.57i)10-s − 1.12i·12-s − 1.14i·13-s + 0.871·14-s + (−0.234 + 0.527i)15-s + 0.853·16-s − 1.42i·17-s + 0.572i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.405 + 0.914i$
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: \((2,\ 1815,\ (\ :1/2),\ -0.405 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.738756127\)
\(L(\frac12)\) \(\approx\) \(1.738756127\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 - 0.906i)T \)
11 \( 1 \)
good2 \( 1 + 2.42iT - 2T^{2} \)
7 \( 1 - 1.34iT - 7T^{2} \)
13 \( 1 + 4.12iT - 13T^{2} \)
17 \( 1 + 5.87iT - 17T^{2} \)
19 \( 1 + 3.51T + 19T^{2} \)
23 \( 1 - 6.37iT - 23T^{2} \)
29 \( 1 - 6.39T + 29T^{2} \)
31 \( 1 - 9.40T + 31T^{2} \)
37 \( 1 + 7.01iT - 37T^{2} \)
41 \( 1 - 8.63T + 41T^{2} \)
43 \( 1 + 5.31iT - 43T^{2} \)
47 \( 1 + 3.74iT - 47T^{2} \)
53 \( 1 + 6.41iT - 53T^{2} \)
59 \( 1 + 8.97T + 59T^{2} \)
61 \( 1 - 6.52T + 61T^{2} \)
67 \( 1 + 8.83iT - 67T^{2} \)
71 \( 1 + 7.17T + 71T^{2} \)
73 \( 1 - 4.08iT - 73T^{2} \)
79 \( 1 - 9.97T + 79T^{2} \)
83 \( 1 + 3.06iT - 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 2.55iT - 97T^{2} \)
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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

−9.280871076772504299280110725067, −8.824843484624146446603671555683, −7.67941405295005021604574697927, −6.37827044276112246349712146866, −5.40560191259378541342082854682, −4.80638842545806918524564310428, −3.62996349516787796155879837497, −2.78494449304593057071298686417, −2.27028940292004398600784340909, −0.78362258930882130011551753291, 1.10706008708979422574109632496, 2.45943143301129218108558798307, 4.37735474305161971930920846137, 4.62901045697279216251594512617, 6.08517858331347031389114851099, 6.26550449103675051490325869192, 6.87454397890461607237945210844, 7.963176057689571787370178631150, 8.489508516274883231783941564900, 9.068273429138278793728805533127

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