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Properties

Label 2-855-95.54-c1-0-46
Degree $2$
Conductor $855$
Sign $-0.949 - 0.315i$
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  + (1.72 − 2.05i)2-s + (−0.906 − 5.13i)4-s + (−1.11 − 1.93i)5-s + (2.41 − 1.39i)7-s + (−7.48 − 4.32i)8-s + (−5.91 − 1.04i)10-s + (−1.18 + 2.06i)11-s + (−0.0138 + 0.0380i)13-s + (1.29 − 7.36i)14-s + (−12.0 + 4.37i)16-s + (−1.16 + 1.39i)17-s + (4.07 − 1.54i)19-s + (−8.93 + 7.50i)20-s + (2.18 + 6.00i)22-s + (2.50 − 0.441i)23-s + ⋯
L(s)  = 1  + (1.22 − 1.45i)2-s + (−0.453 − 2.56i)4-s + (−0.500 − 0.865i)5-s + (0.910 − 0.525i)7-s + (−2.64 − 1.52i)8-s + (−1.87 − 0.329i)10-s + (−0.358 + 0.621i)11-s + (−0.00384 + 0.0105i)13-s + (0.347 − 1.96i)14-s + (−3.00 + 1.09i)16-s + (−0.283 + 0.337i)17-s + (0.935 − 0.354i)19-s + (−1.99 + 1.67i)20-s + (0.466 + 1.28i)22-s + (0.522 − 0.0921i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.949 - 0.315i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.949 - 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418409 + 2.58774i\)
\(L(\frac12)\) \(\approx\) \(0.418409 + 2.58774i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 + 1.93i)T \)
19 \( 1 + (-4.07 + 1.54i)T \)
good2 \( 1 + (-1.72 + 2.05i)T + (-0.347 - 1.96i)T^{2} \)
7 \( 1 + (-2.41 + 1.39i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.18 - 2.06i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0138 - 0.0380i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.16 - 1.39i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-2.50 + 0.441i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.25 + 1.89i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.44 - 2.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.227iT - 37T^{2} \)
41 \( 1 + (-7.55 + 2.74i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (5.05 + 0.891i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (7.11 + 8.48i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (5.62 - 0.992i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-8.89 - 7.46i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.795 - 4.51i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.11 - 3.71i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.34 + 7.65i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (1.45 + 3.99i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-10.9 + 3.97i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-3.87 + 2.23i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (14.4 + 5.26i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (3.59 - 4.28i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.04903266575078036736846515966, −9.182591215221824503172996200843, −8.150833929539226699829579982730, −7.02792710411410725434095808447, −5.59397968000081899828263803688, −4.81188690179554376347121344126, −4.36679560488454779911680515032, −3.29559136916692528797321469352, −1.96638060758754778737708486401, −0.909611589157329687457922902646, 2.71008079015221030362510678103, 3.54348936891347302703746237946, 4.68069265854208447216657413565, 5.38391940515258835200152038994, 6.29554419609384596923626081565, 7.06694457083938854590236568661, 8.004972048177289265009663521489, 8.260048380365594531492365882026, 9.555398107365353354306263385379, 11.13291450278183690305570943362

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