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Properties

Label 2-855-19.7-c1-0-7
Degree $2$
Conductor $855$
Sign $-0.998 + 0.0548i$
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  + (0.898 + 1.55i)2-s + (−0.613 + 1.06i)4-s + (0.5 + 0.866i)5-s − 3.41·7-s + 1.38·8-s + (−0.898 + 1.55i)10-s − 4.35·11-s + (−2.49 + 4.31i)13-s + (−3.06 − 5.31i)14-s + (2.47 + 4.28i)16-s + (0.0290 + 0.0503i)17-s + (−1.10 + 4.21i)19-s − 1.22·20-s + (−3.91 − 6.77i)22-s + (−0.216 + 0.374i)23-s + ⋯
L(s)  = 1  + (0.635 + 1.10i)2-s + (−0.306 + 0.531i)4-s + (0.223 + 0.387i)5-s − 1.29·7-s + 0.490·8-s + (−0.284 + 0.491i)10-s − 1.31·11-s + (−0.690 + 1.19i)13-s + (−0.819 − 1.41i)14-s + (0.618 + 1.07i)16-s + (0.00704 + 0.0122i)17-s + (−0.253 + 0.967i)19-s − 0.274·20-s + (−0.834 − 1.44i)22-s + (−0.0450 + 0.0780i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.998 + 0.0548i$
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: \((2,\ 855,\ (\ :1/2),\ -0.998 + 0.0548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0374495 - 1.36369i\)
\(L(\frac12)\) \(\approx\) \(0.0374495 - 1.36369i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 - 4.21i)T \)
good2 \( 1 + (-0.898 - 1.55i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 4.35T + 11T^{2} \)
13 \( 1 + (2.49 - 4.31i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.0290 - 0.0503i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.216 - 0.374i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.74 - 4.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.592T + 31T^{2} \)
37 \( 1 - 6.62T + 37T^{2} \)
41 \( 1 + (0.0818 + 0.141i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.00 - 5.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.54 + 9.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.69 + 4.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.72 - 2.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.15 - 3.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.39 + 5.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.20 + 7.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.84 - 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.64 + 6.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.1T + 83T^{2} \)
89 \( 1 + (-5.59 + 9.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.47 + 7.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.33307837197476417134053089562, −9.899556412472129349958502147465, −8.806442041362037426767607146094, −7.62080417593514495720891050700, −7.07560811456791093113282071508, −6.23755473127502061369974313332, −5.59488730899645383153009666732, −4.55927306454348954358630932781, −3.46596519991628705382060849462, −2.19511695960160461878624936598, 0.49749612452367855932151625290, 2.49034725385201906287076988658, 2.90999060132347141671685710200, 4.14207780868095548677795508154, 5.14160611345177524809861215448, 5.92491306315890208626411376933, 7.26164684947170319035011267416, 7.993602930831949873777158941302, 9.294987820557035928877354779514, 10.01477459664453562090817081448

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