Properties

Label 2-855-95.44-c1-0-19
Degree $2$
Conductor $855$
Sign $0.568 + 0.822i$
Analytic cond. $6.82720$
Root an. cond. $2.61289$
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  + (−1.04 − 1.24i)2-s + (−0.107 + 0.611i)4-s + (2.12 − 0.705i)5-s + (1.93 + 1.11i)7-s + (−1.93 + 1.11i)8-s + (−3.08 − 1.89i)10-s + (2.82 + 4.88i)11-s + (−1.60 − 4.41i)13-s + (−0.627 − 3.55i)14-s + (4.56 + 1.66i)16-s + (−0.505 − 0.601i)17-s + (2.63 + 3.47i)19-s + (0.202 + 1.37i)20-s + (3.12 − 8.58i)22-s + (5.05 + 0.890i)23-s + ⋯
L(s)  = 1  + (−0.735 − 0.877i)2-s + (−0.0539 + 0.305i)4-s + (0.948 − 0.315i)5-s + (0.730 + 0.421i)7-s + (−0.683 + 0.394i)8-s + (−0.975 − 0.600i)10-s + (0.850 + 1.47i)11-s + (−0.445 − 1.22i)13-s + (−0.167 − 0.951i)14-s + (1.14 + 0.415i)16-s + (−0.122 − 0.146i)17-s + (0.603 + 0.797i)19-s + (0.0453 + 0.307i)20-s + (0.666 − 1.83i)22-s + (1.05 + 0.185i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.568 + 0.822i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.568 + 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23610 - 0.648441i\)
\(L(\frac12)\) \(\approx\) \(1.23610 - 0.648441i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.12 + 0.705i)T \)
19 \( 1 + (-2.63 - 3.47i)T \)
good2 \( 1 + (1.04 + 1.24i)T + (-0.347 + 1.96i)T^{2} \)
7 \( 1 + (-1.93 - 1.11i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.82 - 4.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.60 + 4.41i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.505 + 0.601i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (-5.05 - 0.890i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-2.32 - 1.95i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (4.05 - 7.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.985iT - 37T^{2} \)
41 \( 1 + (-1.33 - 0.484i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.49 + 0.264i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.617 - 0.735i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (4.34 + 0.766i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-7.56 + 6.34i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.363 + 2.06i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.08 - 2.48i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.24 + 7.06i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-5.18 + 14.2i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (-1.25 - 0.458i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (6.51 + 3.76i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.19 - 1.16i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-8.20 - 9.77i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.02282069210267817512251803912, −9.376779878349954230579020094676, −8.744211176548132847808927732142, −7.73910408178956541241874160278, −6.62087171023099359395898334718, −5.42459968556465252230927826896, −4.90729166718299853327113675138, −3.15995374652500352998754843882, −2.01544517849142661390654092972, −1.27345145443751765146952403585, 1.08849147185354809027184760750, 2.69074318300692935693934996754, 4.00538639983583070795255960180, 5.33078044257250343479952688049, 6.27527085311544188985603171202, 6.88360191324502469738289914675, 7.67960288982385084893792953874, 8.841698344164781018453460162635, 9.102839067680749867743416029558, 10.00454634673303690121830842217

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