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Properties

Label 2-855-5.4-c1-0-8
Degree $2$
Conductor $855$
Sign $0.929 + 0.369i$
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  − 2.41i·2-s − 3.85·4-s + (2.07 + 0.826i)5-s + 3.18i·7-s + 4.49i·8-s + (2 − 5.02i)10-s − 4.15·11-s + 2.07i·13-s + 7.71·14-s + 3.15·16-s + 5.79i·17-s + 19-s + (−8.01 − 3.18i)20-s + 10.0i·22-s + 2.60i·23-s + ⋯
L(s)  = 1  − 1.71i·2-s − 1.92·4-s + (0.929 + 0.369i)5-s + 1.20i·7-s + 1.58i·8-s + (0.632 − 1.58i)10-s − 1.25·11-s + 0.574i·13-s + 2.06·14-s + 0.788·16-s + 1.40i·17-s + 0.229·19-s + (−1.79 − 0.712i)20-s + 2.14i·22-s + 0.543i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.929 + 0.369i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28859 - 0.246885i\)
\(L(\frac12)\) \(\approx\) \(1.28859 - 0.246885i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 - 0.826i)T \)
19 \( 1 - T \)
good2 \( 1 + 2.41iT - 2T^{2} \)
7 \( 1 - 3.18iT - 7T^{2} \)
11 \( 1 + 4.15T + 11T^{2} \)
13 \( 1 - 2.07iT - 13T^{2} \)
17 \( 1 - 5.79iT - 17T^{2} \)
23 \( 1 - 2.60iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 2.59T + 31T^{2} \)
37 \( 1 + 4.30iT - 37T^{2} \)
41 \( 1 - 0.599T + 41T^{2} \)
43 \( 1 - 3.18iT - 43T^{2} \)
47 \( 1 - 11.7iT - 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 + 1.71T + 59T^{2} \)
61 \( 1 + 8.75T + 61T^{2} \)
67 \( 1 + 4.76iT - 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + 2.72iT - 73T^{2} \)
79 \( 1 - 1.40T + 79T^{2} \)
83 \( 1 - 7.07iT - 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 + 2.07iT - 97T^{2} \)
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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

−10.32183615496950039706167111469, −9.519033919907330899184181070219, −8.888721464907335097805666701353, −7.958900053104761522206995142892, −6.37423966055257809890990921645, −5.53492046650656864865377216179, −4.59587082209430712492227982087, −3.19544824594187826736925968366, −2.45633807964518455770823017436, −1.63804014048652847323990572895, 0.64763631516942824428922256579, 2.81015749332306697732178237401, 4.53401789975015927981378414551, 5.07795794433856567794343532271, 5.92858558804134154164045246438, 6.87890290265266102308653110201, 7.51834213519211023958265080036, 8.290849281218151819659701589102, 9.143229251726419436801851306446, 10.13604708933689707918974912966

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