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Properties

Label 2-855-19.11-c1-0-18
Degree $2$
Conductor $855$
Sign $-0.0178 + 0.999i$
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  + (1.00 − 1.74i)2-s + (−1.02 − 1.78i)4-s + (−0.5 + 0.866i)5-s + 1.66·7-s − 0.118·8-s + (1.00 + 1.74i)10-s + 0.745·11-s + (0.269 + 0.467i)13-s + (1.67 − 2.90i)14-s + (1.93 − 3.35i)16-s + (0.705 − 1.22i)17-s + (4.17 − 1.25i)19-s + 2.05·20-s + (0.750 − 1.30i)22-s + (0.437 + 0.757i)23-s + ⋯
L(s)  = 1  + (0.712 − 1.23i)2-s + (−0.514 − 0.891i)4-s + (−0.223 + 0.387i)5-s + 0.629·7-s − 0.0419·8-s + (0.318 + 0.551i)10-s + 0.224·11-s + (0.0748 + 0.129i)13-s + (0.448 − 0.776i)14-s + (0.484 − 0.839i)16-s + (0.171 − 0.296i)17-s + (0.957 − 0.288i)19-s + 0.460·20-s + (0.160 − 0.277i)22-s + (0.0912 + 0.157i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0178 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0178 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.0178 + 0.999i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.0178 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75376 - 1.78540i\)
\(L(\frac12)\) \(\approx\) \(1.75376 - 1.78540i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.17 + 1.25i)T \)
good2 \( 1 + (-1.00 + 1.74i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 - 1.66T + 7T^{2} \)
11 \( 1 - 0.745T + 11T^{2} \)
13 \( 1 + (-0.269 - 0.467i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.705 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.437 - 0.757i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.70 + 6.41i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.02T + 31T^{2} \)
37 \( 1 - 7.84T + 37T^{2} \)
41 \( 1 + (4.36 - 7.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.93 - 5.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.29 + 5.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.85 + 6.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.45 - 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.60 - 4.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.443 + 0.768i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.41 - 2.44i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.524 - 0.908i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.88 - 4.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.61T + 83T^{2} \)
89 \( 1 + (-0.464 - 0.804i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.00920 - 0.0159i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.05448735107925307175544728588, −9.604083970352481304287002731682, −8.208700512169066359768239542052, −7.51090985322960826972348581023, −6.34793879309811466255410301404, −5.13758773938620101454986288476, −4.39755434427765790382425289560, −3.38922836394982558912071446640, −2.48772355851593536418336953401, −1.21845855920114986647455245277, 1.47338999678558671274577347174, 3.37760894501137322310348418602, 4.45088220007358908522167802246, 5.15706037558898718159795606289, 5.95717384536677109986031697983, 6.91326421474076092231929662076, 7.75993440943775458134101639589, 8.311002375676469339923746585895, 9.307673570703587272684293230580, 10.41149178779008757586509304303

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