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Properties

Label 2-855-19.7-c1-0-27
Degree $2$
Conductor $855$
Sign $0.378 + 0.925i$
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  + (0.464 + 0.804i)2-s + (0.568 − 0.984i)4-s + (−0.5 − 0.866i)5-s − 3.66·7-s + 2.91·8-s + (0.464 − 0.804i)10-s + 1.41·11-s + (1.38 − 2.39i)13-s + (−1.70 − 2.94i)14-s + (0.216 + 0.374i)16-s + (−1.62 − 2.81i)17-s + (−4.34 + 0.336i)19-s − 1.13·20-s + (0.658 + 1.14i)22-s + (4.21 − 7.29i)23-s + ⋯
L(s)  = 1  + (0.328 + 0.568i)2-s + (0.284 − 0.492i)4-s + (−0.223 − 0.387i)5-s − 1.38·7-s + 1.03·8-s + (0.146 − 0.254i)10-s + 0.427·11-s + (0.383 − 0.663i)13-s + (−0.454 − 0.787i)14-s + (0.0540 + 0.0936i)16-s + (−0.393 − 0.681i)17-s + (−0.997 + 0.0772i)19-s − 0.254·20-s + (0.140 + 0.243i)22-s + (0.878 − 1.52i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.378 + 0.925i$
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: \((2,\ 855,\ (\ :1/2),\ 0.378 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24638 - 0.836996i\)
\(L(\frac12)\) \(\approx\) \(1.24638 - 0.836996i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.34 - 0.336i)T \)
good2 \( 1 + (-0.464 - 0.804i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 3.66T + 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 + (-1.38 + 2.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.62 + 2.81i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.21 + 7.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.917 + 1.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.85T + 31T^{2} \)
37 \( 1 + 9.00T + 37T^{2} \)
41 \( 1 + (1.82 + 3.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.41 + 7.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.37 + 9.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.56 - 6.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.65 - 4.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.64 + 2.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.32 - 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.28 - 12.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.964 + 1.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.25 - 10.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.46T + 83T^{2} \)
89 \( 1 + (-5.27 + 9.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.311 - 0.539i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.23463280604152393878024758000, −9.057018062152816905589707073645, −8.400710236116696866341783630911, −6.98914659197292000506822286753, −6.66984244171689819693616650052, −5.75004498075543340894235536762, −4.79382606467133991318677585134, −3.74699453345249183204617524672, −2.44871443536525424311067080586, −0.64450585148328625532375558497, 1.78018175126674702442617962390, 3.13325629599889226894999845028, 3.63117959570434677938916164646, 4.67834738959159443004043379284, 6.36923153875030141332027824735, 6.65727745182226727933989318324, 7.71165942006154202944068225165, 8.760564848261809028309590705407, 9.588574485516309240824385961100, 10.54800633062986745249294600504

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