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Properties

Label 2-855-19.7-c1-0-2
Degree $2$
Conductor $855$
Sign $0.615 - 0.788i$
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  + (−1.34 − 2.33i)2-s + (−2.62 + 4.54i)4-s + (0.5 + 0.866i)5-s − 0.797·7-s + 8.73·8-s + (1.34 − 2.33i)10-s − 2.59·11-s + (1.39 − 2.42i)13-s + (1.07 + 1.85i)14-s + (−6.50 − 11.2i)16-s + (−2.88 − 4.99i)17-s + (2.45 + 3.60i)19-s − 5.24·20-s + (3.48 + 6.04i)22-s + (−1.55 + 2.68i)23-s + ⋯
L(s)  = 1  + (−0.951 − 1.64i)2-s + (−1.31 + 2.27i)4-s + (0.223 + 0.387i)5-s − 0.301·7-s + 3.08·8-s + (0.425 − 0.737i)10-s − 0.781·11-s + (0.387 − 0.671i)13-s + (0.286 + 0.496i)14-s + (−1.62 − 2.81i)16-s + (−0.700 − 1.21i)17-s + (0.562 + 0.826i)19-s − 1.17·20-s + (0.743 + 1.28i)22-s + (−0.323 + 0.560i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.236553 + 0.115448i\)
\(L(\frac12)\) \(\approx\) \(0.236553 + 0.115448i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 - 3.60i)T \)
good2 \( 1 + (1.34 + 2.33i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 0.797T + 7T^{2} \)
11 \( 1 + 2.59T + 11T^{2} \)
13 \( 1 + (-1.39 + 2.42i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.88 + 4.99i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.55 - 2.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.39 - 4.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.48T + 31T^{2} \)
37 \( 1 - 7.69T + 37T^{2} \)
41 \( 1 + (-3.69 - 6.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.39 - 2.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.53 - 9.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.43 - 7.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.540 - 0.935i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.03 - 3.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.88 - 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.98 + 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.25 + 7.36i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.24 + 5.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.79T + 83T^{2} \)
89 \( 1 + (6.67 - 11.5i)T + (-44.5 - 77.0i)T^{2} \)
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

−10.34116881028450907917776304720, −9.556577141408212470664117821540, −9.088351029025654123162585076802, −7.86242028326405427129666496038, −7.45042931497203957003952776002, −5.85769073522034600479660005671, −4.58359337660004449845468638798, −3.31492392591159664152133541566, −2.75472719549048960286671918024, −1.45412320360186356602744331257, 0.18013229128337913096186245505, 1.89595407148797401177457030229, 4.09733131179694296434640074628, 5.13546144782887627758283129465, 5.94971220871469032924727867555, 6.66467236789984922286106930298, 7.54170675149234265396688706700, 8.346792249042652406884572859944, 9.021913442403471577975414586617, 9.663022687502020795338160834918

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