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Properties

Label 2-1815-5.4-c1-0-48
Degree $2$
Conductor $1815$
Sign $-0.295 - 0.955i$
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  + 0.803i·2-s + i·3-s + 1.35·4-s + (2.13 − 0.660i)5-s − 0.803·6-s − 0.508i·7-s + 2.69i·8-s − 9-s + (0.530 + 1.71i)10-s + 1.35i·12-s + 5.13i·13-s + 0.408·14-s + (0.660 + 2.13i)15-s + 0.544·16-s + 3.34i·17-s − 0.803i·18-s + ⋯
L(s)  = 1  + 0.568i·2-s + 0.577i·3-s + 0.677·4-s + (0.955 − 0.295i)5-s − 0.327·6-s − 0.192i·7-s + 0.952i·8-s − 0.333·9-s + (0.167 + 0.542i)10-s + 0.391i·12-s + 1.42i·13-s + 0.109·14-s + (0.170 + 0.551i)15-s + 0.136·16-s + 0.811i·17-s − 0.189i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.295 - 0.955i$
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: \((2,\ 1815,\ (\ :1/2),\ -0.295 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.495230079\)
\(L(\frac12)\) \(\approx\) \(2.495230079\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 + 0.660i)T \)
11 \( 1 \)
good2 \( 1 - 0.803iT - 2T^{2} \)
7 \( 1 + 0.508iT - 7T^{2} \)
13 \( 1 - 5.13iT - 13T^{2} \)
17 \( 1 - 3.34iT - 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 + 2.91iT - 23T^{2} \)
29 \( 1 - 0.392T + 29T^{2} \)
31 \( 1 + 6.35T + 31T^{2} \)
37 \( 1 - 4.45iT - 37T^{2} \)
41 \( 1 + 2.79T + 41T^{2} \)
43 \( 1 - 3.05iT - 43T^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 - 9.80iT - 53T^{2} \)
59 \( 1 - 3.02T + 59T^{2} \)
61 \( 1 - 1.05T + 61T^{2} \)
67 \( 1 - 5.31iT - 67T^{2} \)
71 \( 1 - 4.09T + 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 - 9.84T + 89T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−9.329452161073638524842973417361, −8.824717402124548603968353980685, −7.933646518461818867305673423285, −6.87780070756949448226935277557, −6.37530124797747590318775747603, −5.55562780413608955864518755644, −4.78818169513830418549138987282, −3.75052296495364331439655599831, −2.44934974661202874403038368899, −1.60376231858569153548490583898, 0.914975711485402767138664232191, 2.03925974076909483328889642286, 2.78073025445871449857105261423, 3.56411031506548849878080075874, 5.30946780970747394320317886156, 5.74423151058622694426565069551, 6.73390733597162670677560605784, 7.32122686903708899521224639381, 8.136210639095277856489449419897, 9.318154497968603958341911654710

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