Properties

Label 2-855-95.54-c1-0-37
Degree $2$
Conductor $855$
Sign $-0.306 + 0.951i$
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  + (1.04 − 1.24i)2-s + (−0.107 − 0.611i)4-s + (−2.23 − 0.0626i)5-s + (−1.93 + 1.11i)7-s + (1.93 + 1.11i)8-s + (−2.40 + 2.70i)10-s + (2.82 − 4.88i)11-s + (1.60 − 4.41i)13-s + (−0.627 + 3.55i)14-s + (4.56 − 1.66i)16-s + (0.505 − 0.601i)17-s + (2.63 − 3.47i)19-s + (0.202 + 1.37i)20-s + (−3.12 − 8.58i)22-s + (−5.05 + 0.890i)23-s + ⋯
L(s)  = 1  + (0.735 − 0.877i)2-s + (−0.0539 − 0.305i)4-s + (−0.999 − 0.0280i)5-s + (−0.730 + 0.421i)7-s + (0.683 + 0.394i)8-s + (−0.760 + 0.856i)10-s + (0.850 − 1.47i)11-s + (0.445 − 1.22i)13-s + (−0.167 + 0.951i)14-s + (1.14 − 0.415i)16-s + (0.122 − 0.146i)17-s + (0.603 − 0.797i)19-s + (0.0453 + 0.307i)20-s + (−0.666 − 1.83i)22-s + (−1.05 + 0.185i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.306 + 0.951i$
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: \((2,\ 855,\ (\ :1/2),\ -0.306 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09621 - 1.50472i\)
\(L(\frac12)\) \(\approx\) \(1.09621 - 1.50472i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0626i)T \)
19 \( 1 + (-2.63 + 3.47i)T \)
good2 \( 1 + (-1.04 + 1.24i)T + (-0.347 - 1.96i)T^{2} \)
7 \( 1 + (1.93 - 1.11i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.82 + 4.88i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.60 + 4.41i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.505 + 0.601i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (5.05 - 0.890i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.32 + 1.95i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (4.05 + 7.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.985iT - 37T^{2} \)
41 \( 1 + (-1.33 + 0.484i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.49 + 0.264i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.617 - 0.735i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-4.34 + 0.766i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-7.56 - 6.34i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.363 - 2.06i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.08 - 2.48i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.24 - 7.06i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (5.18 + 14.2i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-1.25 + 0.458i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.51 + 3.76i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.19 + 1.16i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (8.20 - 9.77i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.21931939523966735258406820534, −9.075003624994394679055693506951, −8.244808490497018876032082514365, −7.50289664956426488983751072769, −6.20055837849788259580732474796, −5.39696179371719934527050004898, −4.07507635670539974300729682027, −3.44369850912517660970844283924, −2.73917662185100911475193444644, −0.78105897297501602000831559675, 1.56972126689182585432782492794, 3.68983758536425002380555830667, 4.09307336671654824038970848674, 5.04111728732224224135808294990, 6.34060501599948889035715014083, 6.91606132691899194439482485188, 7.44572153800093411998450763266, 8.552582400095388786182424856366, 9.666447002219002146146417678868, 10.28775338452655619236637972132

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