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Properties

Label 2-855-19.7-c1-0-30
Degree $2$
Conductor $855$
Sign $-0.999 - 0.0417i$
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  + (−0.0850 − 0.147i)2-s + (0.985 − 1.70i)4-s + (−0.5 − 0.866i)5-s − 0.218·7-s − 0.675·8-s + (−0.0850 + 0.147i)10-s − 5.33·11-s + (−3.01 + 5.22i)13-s + (0.0185 + 0.0321i)14-s + (−1.91 − 3.31i)16-s + (−3.27 − 5.67i)17-s + (1.50 − 4.09i)19-s − 1.97·20-s + (0.453 + 0.785i)22-s + (−1.90 + 3.30i)23-s + ⋯
L(s)  = 1  + (−0.0601 − 0.104i)2-s + (0.492 − 0.853i)4-s + (−0.223 − 0.387i)5-s − 0.0825·7-s − 0.238·8-s + (−0.0268 + 0.0465i)10-s − 1.60·11-s + (−0.836 + 1.44i)13-s + (0.00496 + 0.00860i)14-s + (−0.478 − 0.828i)16-s + (−0.794 − 1.37i)17-s + (0.345 − 0.938i)19-s − 0.440·20-s + (0.0966 + 0.167i)22-s + (−0.397 + 0.689i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.999 - 0.0417i$
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: \((2,\ 855,\ (\ :1/2),\ -0.999 - 0.0417i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0109774 + 0.525180i\)
\(L(\frac12)\) \(\approx\) \(0.0109774 + 0.525180i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 + 4.09i)T \)
good2 \( 1 + (0.0850 + 0.147i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 0.218T + 7T^{2} \)
11 \( 1 + 5.33T + 11T^{2} \)
13 \( 1 + (3.01 - 5.22i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.27 + 5.67i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.90 - 3.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.525 - 0.909i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.65T + 31T^{2} \)
37 \( 1 - 3.10T + 37T^{2} \)
41 \( 1 + (-1.75 - 3.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.39 + 2.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.26 - 7.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.57 + 4.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.95 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.25 + 3.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.30 + 5.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.12 + 5.40i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.31 + 5.74i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.80 + 3.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.39T + 83T^{2} \)
89 \( 1 + (-4.79 + 8.29i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.95 - 13.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.475316234330693298202105216389, −9.396432102764578300445376587335, −7.922514414284109371076503585110, −7.15281195759102844469482253675, −6.35135710928953516375927986792, −4.99593935080146238313243077961, −4.80240783376486732197434399234, −2.89566678659815585165542933568, −2.00153650078747622280859994840, −0.23052204136007210765018934306, 2.33167536831916053544740884252, 3.05951621739745766048117164094, 4.17565336875961817511639111046, 5.46005167355968461026038065524, 6.33966991371218087761415779407, 7.45848937011203560645991672777, 7.970049092512430033699094037184, 8.530879927357789757609281506348, 10.16551606832545567453363437802, 10.41491296137533954227531895999

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