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Properties

Label 2-855-95.7-c0-0-0
Degree $2$
Conductor $855$
Sign $-0.566 - 0.823i$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.965 − 0.258i)5-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)10-s + 1.41·11-s + (0.366 + 1.36i)13-s + (−0.5 − 0.866i)16-s + (−0.258 + 0.965i)17-s + (−0.866 + 0.5i)19-s + (−0.366 + 1.36i)22-s + (0.866 + 0.499i)25-s − 1.41·26-s − 31-s + (−0.866 − 0.499i)34-s + (−1 − i)37-s + (−0.258 − 0.965i)38-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.965 − 0.258i)5-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)10-s + 1.41·11-s + (0.366 + 1.36i)13-s + (−0.5 − 0.866i)16-s + (−0.258 + 0.965i)17-s + (−0.866 + 0.5i)19-s + (−0.366 + 1.36i)22-s + (0.866 + 0.499i)25-s − 1.41·26-s − 31-s + (−0.866 − 0.499i)34-s + (−1 − i)37-s + (−0.258 − 0.965i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.566 - 0.823i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ -0.566 - 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7828962650\)
\(L(\frac12)\) \(\approx\) \(0.7828962650\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.866 - 0.5i)T^{2} \)
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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

−10.90240937978299165744042890078, −9.377567106693895898734123283517, −8.759449982680199547128784370041, −8.191468868949746426395312558979, −7.10682102771022303400301754039, −6.62944490753616164244958329847, −5.73593292682830566395997303632, −4.27767710724532506320865810744, −3.71578746602802863053047254344, −1.87486847718341962188383069419, 0.906812276728698272074293479782, 2.53043565238001305518755186535, 3.48975052042451630745153695327, 4.29003292911573883334223970645, 5.77302577639904556740014710629, 6.78324204572897541609898096164, 7.48964302535801901011244833597, 8.759632363707424402579495963049, 9.212743098825318142524975929829, 10.45621039257552304345687369106

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