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Properties

Label 2-1815-5.4-c1-0-63
Degree $2$
Conductor $1815$
Sign $0.925 - 0.379i$
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  + 1.19i·2-s − i·3-s + 0.570·4-s + (0.849 + 2.06i)5-s + 1.19·6-s − 3.76i·7-s + 3.07i·8-s − 9-s + (−2.47 + 1.01i)10-s − 0.570i·12-s − 1.11i·13-s + 4.49·14-s + (2.06 − 0.849i)15-s − 2.53·16-s − 5.57i·17-s − 1.19i·18-s + ⋯
L(s)  = 1  + 0.845i·2-s − 0.577i·3-s + 0.285·4-s + (0.379 + 0.925i)5-s + 0.488·6-s − 1.42i·7-s + 1.08i·8-s − 0.333·9-s + (−0.782 + 0.321i)10-s − 0.164i·12-s − 0.308i·13-s + 1.20·14-s + (0.534 − 0.219i)15-s − 0.633·16-s − 1.35i·17-s − 0.281i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.925 - 0.379i$
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: \((2,\ 1815,\ (\ :1/2),\ 0.925 - 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.226982601\)
\(L(\frac12)\) \(\approx\) \(2.226982601\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.849 - 2.06i)T \)
11 \( 1 \)
good2 \( 1 - 1.19iT - 2T^{2} \)
7 \( 1 + 3.76iT - 7T^{2} \)
13 \( 1 + 1.11iT - 13T^{2} \)
17 \( 1 + 5.57iT - 17T^{2} \)
19 \( 1 - 1.37T + 19T^{2} \)
23 \( 1 - 2.35iT - 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 - 9.94T + 31T^{2} \)
37 \( 1 + 4.01iT - 37T^{2} \)
41 \( 1 - 3.94T + 41T^{2} \)
43 \( 1 + 7.78iT - 43T^{2} \)
47 \( 1 - 6.46iT - 47T^{2} \)
53 \( 1 + 4.06iT - 53T^{2} \)
59 \( 1 - 4.28T + 59T^{2} \)
61 \( 1 + 8.32T + 61T^{2} \)
67 \( 1 - 9.15iT - 67T^{2} \)
71 \( 1 + 2.85T + 71T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 - 3.07T + 79T^{2} \)
83 \( 1 - 4.64iT - 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 - 18.1iT - 97T^{2} \)
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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

−9.248017318097301698174730029576, −8.018683696500821554324663993954, −7.57515267655758853554914930822, −6.86643025660946560056818131366, −6.51407398076343578978493140516, −5.54994351058181154428440808634, −4.56593687101636429004810882958, −3.19006558834476002483173166157, −2.44352281785304700340579859085, −0.964119870633139159331298795936, 1.18873450576800984066730409208, 2.28410362318364636809668242752, 3.01223997335402505805313378651, 4.25730538407742827485830249814, 4.94564426973985730387991822214, 6.08034051388803413277480712278, 6.40626913004023796076596886159, 8.079076327394718565620687993687, 8.609412517338140582298616865288, 9.343886910938670191083236621840

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