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Properties

Label 2-855-19.7-c1-0-25
Degree $2$
Conductor $855$
Sign $-0.0977 + 0.995i$
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  + (−0.207 − 0.358i)2-s + (0.914 − 1.58i)4-s + (−0.5 − 0.866i)5-s + 1.82·7-s − 1.58·8-s + (−0.207 + 0.358i)10-s + 2.82·11-s + (0.914 − 1.58i)13-s + (−0.378 − 0.655i)14-s + (−1.49 − 2.59i)16-s + (−0.585 − 1.01i)17-s + (4 + 1.73i)19-s − 1.82·20-s + (−0.585 − 1.01i)22-s + (−0.414 + 0.717i)23-s + ⋯
L(s)  = 1  + (−0.146 − 0.253i)2-s + (0.457 − 0.791i)4-s + (−0.223 − 0.387i)5-s + 0.691·7-s − 0.560·8-s + (−0.0654 + 0.113i)10-s + 0.852·11-s + (0.253 − 0.439i)13-s + (−0.101 − 0.175i)14-s + (−0.374 − 0.649i)16-s + (−0.142 − 0.246i)17-s + (0.917 + 0.397i)19-s − 0.408·20-s + (−0.124 − 0.216i)22-s + (−0.0863 + 0.149i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.0977 + 0.995i$
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: \((2,\ 855,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11763 - 1.23277i\)
\(L(\frac12)\) \(\approx\) \(1.11763 - 1.23277i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 - 1.73i)T \)
good2 \( 1 + (0.207 + 0.358i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 - 1.82T + 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + (-0.914 + 1.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.585 + 1.01i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.414 - 0.717i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.82 + 8.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 2.17T + 37T^{2} \)
41 \( 1 + (-1.41 - 2.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.91 + 6.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.58 - 2.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.15 + 7.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.74 - 4.75i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5 - 8.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.74 + 8.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.67 - 2.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (-6.24 + 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.947722280057485080315481949949, −9.258567081997665989987758629663, −8.326299055699953988930225555920, −7.44159217249037496392029981502, −6.40476831805804803909153137550, −5.55215288811809189456164127551, −4.66718936476540772382927057986, −3.43197614292778025536600580339, −1.99465804068466328322066550493, −0.918218696833690515175329225242, 1.65939686926094256625508042628, 3.05356233117805522530622085727, 3.92471164930328643949208812155, 5.07077938489908334972766734758, 6.40437767461734359444151609195, 6.98640162379484144943588917881, 7.81120901884500628590531389722, 8.639325829248419407696360072546, 9.307887442883313096730002718773, 10.59086765557681449112254109521

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