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Properties

Label 2-855-15.2-c1-0-17
Degree $2$
Conductor $855$
Sign $0.961 - 0.274i$
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  + (0.469 − 0.469i)2-s + 1.55i·4-s + (1.73 − 1.40i)5-s + (−0.563 − 0.563i)7-s + (1.67 + 1.67i)8-s + (0.155 − 1.47i)10-s + 5.21i·11-s + (1.60 − 1.60i)13-s − 0.529·14-s − 1.54·16-s + (2.60 − 2.60i)17-s + i·19-s + (2.19 + 2.70i)20-s + (2.44 + 2.44i)22-s + (5.91 + 5.91i)23-s + ⋯
L(s)  = 1  + (0.332 − 0.332i)2-s + 0.779i·4-s + (0.777 − 0.629i)5-s + (−0.212 − 0.212i)7-s + (0.590 + 0.590i)8-s + (0.0492 − 0.467i)10-s + 1.57i·11-s + (0.443 − 0.443i)13-s − 0.141·14-s − 0.386·16-s + (0.632 − 0.632i)17-s + 0.229i·19-s + (0.490 + 0.605i)20-s + (0.521 + 0.521i)22-s + (1.23 + 1.23i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.961 - 0.274i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.961 - 0.274i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16562 + 0.303612i\)
\(L(\frac12)\) \(\approx\) \(2.16562 + 0.303612i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 1.40i)T \)
19 \( 1 - iT \)
good2 \( 1 + (-0.469 + 0.469i)T - 2iT^{2} \)
7 \( 1 + (0.563 + 0.563i)T + 7iT^{2} \)
11 \( 1 - 5.21iT - 11T^{2} \)
13 \( 1 + (-1.60 + 1.60i)T - 13iT^{2} \)
17 \( 1 + (-2.60 + 2.60i)T - 17iT^{2} \)
23 \( 1 + (-5.91 - 5.91i)T + 23iT^{2} \)
29 \( 1 + 5.99T + 29T^{2} \)
31 \( 1 - 3.10T + 31T^{2} \)
37 \( 1 + (-3.21 - 3.21i)T + 37iT^{2} \)
41 \( 1 + 1.54iT - 41T^{2} \)
43 \( 1 + (-1.72 + 1.72i)T - 43iT^{2} \)
47 \( 1 + (1.44 - 1.44i)T - 47iT^{2} \)
53 \( 1 + (-1.07 - 1.07i)T + 53iT^{2} \)
59 \( 1 - 8.52T + 59T^{2} \)
61 \( 1 - 5.80T + 61T^{2} \)
67 \( 1 + (9.57 + 9.57i)T + 67iT^{2} \)
71 \( 1 - 9.02iT - 71T^{2} \)
73 \( 1 + (4.66 - 4.66i)T - 73iT^{2} \)
79 \( 1 + 3.66iT - 79T^{2} \)
83 \( 1 + (-7.43 - 7.43i)T + 83iT^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (4.26 + 4.26i)T + 97iT^{2} \)
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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

−10.04753021298775576891479017805, −9.523395768200120826282589606266, −8.586236359431497318744448395038, −7.58368484153170067278291028565, −6.95525897608925559389741706986, −5.55807121807926272569174633400, −4.84105495813749505138223254264, −3.82847893116657902146945761576, −2.69729560632999861970991668841, −1.51169462521869995804464246524, 1.12362102699131422580197264345, 2.59805892845913673304029327293, 3.74044097205132324767996652720, 5.08386680568439675462447964816, 6.02483998375110625465243049352, 6.27641373944912570660188704065, 7.31596566463150040152431510564, 8.632537887190122375859473957078, 9.297226253678622790079587400545, 10.25941997703408864007345444066

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