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Properties

Label 2-855-95.64-c1-0-33
Degree $2$
Conductor $855$
Sign $0.690 + 0.723i$
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  + (0.747 − 0.431i)2-s + (−0.627 + 1.08i)4-s + (0.0476 − 2.23i)5-s − 0.566i·7-s + 2.80i·8-s + (−0.928 − 1.69i)10-s + 1.91·11-s + (−0.168 − 0.0972i)13-s + (−0.244 − 0.423i)14-s + (−0.0438 − 0.0760i)16-s + (4.58 − 2.64i)17-s + (2.36 − 3.65i)19-s + (2.40 + 1.45i)20-s + (1.42 − 0.824i)22-s + (2.92 + 1.68i)23-s + ⋯
L(s)  = 1  + (0.528 − 0.305i)2-s + (−0.313 + 0.543i)4-s + (0.0213 − 0.999i)5-s − 0.214i·7-s + 0.993i·8-s + (−0.293 − 0.534i)10-s + 0.576·11-s + (−0.0467 − 0.0269i)13-s + (−0.0653 − 0.113i)14-s + (−0.0109 − 0.0190i)16-s + (1.11 − 0.642i)17-s + (0.543 − 0.839i)19-s + (0.536 + 0.325i)20-s + (0.304 − 0.175i)22-s + (0.609 + 0.351i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81089 - 0.774793i\)
\(L(\frac12)\) \(\approx\) \(1.81089 - 0.774793i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0476 + 2.23i)T \)
19 \( 1 + (-2.36 + 3.65i)T \)
good2 \( 1 + (-0.747 + 0.431i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + 0.566iT - 7T^{2} \)
11 \( 1 - 1.91T + 11T^{2} \)
13 \( 1 + (0.168 + 0.0972i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.58 + 2.64i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.92 - 1.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.36 + 7.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 0.955iT - 37T^{2} \)
41 \( 1 + (-5.02 - 8.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.27 - 2.46i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.65 + 4.41i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.10 + 4.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.85 + 3.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.75 + 3.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.50 - 2.02i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.59 - 4.49i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.45 + 4.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.31 - 5.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.51iT - 83T^{2} \)
89 \( 1 + (1.68 - 2.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.1 - 7.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.796247458164800074778304073899, −9.357626878390318871792589518630, −8.274771907871089101299146773215, −7.78295782164131559137755784160, −6.53884329952143064202705257797, −5.26178083232976903067068380929, −4.71464612528221052077952227245, −3.74044212783345727943274057268, −2.68658290892472637134242952658, −0.987357862996166031487783090832, 1.37371532421171773173121824289, 3.06493475997924747764611791837, 3.92055776184960951108490788917, 5.09845012852547660866181017840, 5.97684234378048338750123544877, 6.62545049676923803319192391415, 7.50770549437324376365139390054, 8.618381222230851886599178559092, 9.637905711129668643780893700101, 10.26559821522325428227827409889

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