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Properties

Label 2-855-19.7-c1-0-21
Degree $2$
Conductor $855$
Sign $-0.962 + 0.272i$
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  + (−0.832 − 1.44i)2-s + (−0.385 + 0.667i)4-s + (0.5 + 0.866i)5-s − 2.43·7-s − 2.04·8-s + (0.832 − 1.44i)10-s + 5.75·11-s + (0.797 − 1.38i)13-s + (2.02 + 3.51i)14-s + (2.47 + 4.28i)16-s + (−2.99 − 5.18i)17-s + (0.149 − 4.35i)19-s − 0.770·20-s + (−4.78 − 8.29i)22-s + (−0.470 + 0.814i)23-s + ⋯
L(s)  = 1  + (−0.588 − 1.01i)2-s + (−0.192 + 0.333i)4-s + (0.223 + 0.387i)5-s − 0.920·7-s − 0.723·8-s + (0.263 − 0.455i)10-s + 1.73·11-s + (0.221 − 0.383i)13-s + (0.541 + 0.938i)14-s + (0.618 + 1.07i)16-s + (−0.725 − 1.25i)17-s + (0.0342 − 0.999i)19-s − 0.172·20-s + (−1.02 − 1.76i)22-s + (−0.0980 + 0.169i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.962 + 0.272i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.962 + 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115273 - 0.828601i\)
\(L(\frac12)\) \(\approx\) \(0.115273 - 0.828601i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.149 + 4.35i)T \)
good2 \( 1 + (0.832 + 1.44i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 2.43T + 7T^{2} \)
11 \( 1 - 5.75T + 11T^{2} \)
13 \( 1 + (-0.797 + 1.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.99 + 5.18i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.470 - 0.814i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.30 + 2.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + 2.89T + 37T^{2} \)
41 \( 1 + (3.15 + 5.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.26 + 3.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.47 + 7.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.09 - 1.90i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.39 + 9.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.26 + 9.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.504 - 0.874i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.41 - 7.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.12 - 8.87i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.80 + 6.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.11T + 83T^{2} \)
89 \( 1 + (5.55 - 9.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.02 + 3.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.636837030151793933378388933548, −9.343986293075103010911957215849, −8.585059925627704184699044382152, −6.91951889666994480654277295136, −6.61712719514904495702439197614, −5.43722387299105896898049648838, −3.88472679384120401836427720581, −3.07030947619493202354841921007, −1.98599584694246623578588368616, −0.50101185179644213052015808481, 1.54307165594303187994163764905, 3.35626474904445443186868890322, 4.25494252330700824023230794395, 5.85656080214613078406659786619, 6.35920203583620029445137389172, 6.96724301849161533511742614207, 8.120268008720542017168184898610, 8.903947516319117569535886804219, 9.333608382420005169910813054062, 10.21158190727400520614609796735

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