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Properties

Label 2-855-19.11-c1-0-20
Degree $2$
Conductor $855$
Sign $0.769 - 0.638i$
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  + (−0.823 + 1.42i)2-s + (−0.355 − 0.616i)4-s + (0.5 − 0.866i)5-s + 4.47·7-s − 2.12·8-s + (0.823 + 1.42i)10-s + 3.44·11-s + (−1.23 − 2.14i)13-s + (−3.68 + 6.38i)14-s + (2.45 − 4.25i)16-s + (3.81 − 6.60i)17-s + (−3.67 − 2.34i)19-s − 0.711·20-s + (−2.83 + 4.90i)22-s + (1.93 + 3.35i)23-s + ⋯
L(s)  = 1  + (−0.582 + 1.00i)2-s + (−0.177 − 0.308i)4-s + (0.223 − 0.387i)5-s + 1.69·7-s − 0.750·8-s + (0.260 + 0.450i)10-s + 1.03·11-s + (−0.343 − 0.595i)13-s + (−0.985 + 1.70i)14-s + (0.614 − 1.06i)16-s + (0.925 − 1.60i)17-s + (−0.843 − 0.537i)19-s − 0.159·20-s + (−0.604 + 1.04i)22-s + (0.403 + 0.698i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.769 - 0.638i$
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: \((2,\ 855,\ (\ :1/2),\ 0.769 - 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38624 + 0.500038i\)
\(L(\frac12)\) \(\approx\) \(1.38624 + 0.500038i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.67 + 2.34i)T \)
good2 \( 1 + (0.823 - 1.42i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 - 3.44T + 11T^{2} \)
13 \( 1 + (1.23 + 2.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.81 + 6.60i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.93 - 3.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.36 + 7.56i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.422T + 31T^{2} \)
37 \( 1 - 3.90T + 37T^{2} \)
41 \( 1 + (-2.64 + 4.58i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.23 - 2.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.338 + 0.586i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.74 - 9.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.26 - 7.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.10 + 7.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.81 - 8.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.92 - 3.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.39 - 14.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.06 - 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.03T + 83T^{2} \)
89 \( 1 + (1.57 + 2.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.907406765432280298217730529180, −9.163695565900750602408550774829, −8.499155299707580297953426726703, −7.61810861980515769976810694598, −7.22211440157516930787022330203, −5.90128884972096759439816284851, −5.23075302915860307421968353811, −4.21384669714707872364007370184, −2.57592973169373693151326182425, −1.01097008635882094005737397435, 1.47844804734515720309540862342, 1.93986239412817196347089364550, 3.47515590496716097964239144141, 4.48838699979614861611552805756, 5.73910941185327792498786216759, 6.61148975313282825056153767835, 7.84392063545129650469508765247, 8.613285372182793552590966561352, 9.297943964301277764736272606271, 10.40192955961282810700947854843

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