Properties

Label 2-855-5.4-c1-0-5
Degree $2$
Conductor $855$
Sign $-0.894 - 0.447i$
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  + i·2-s + 4-s + (−2 − i)5-s + 2i·7-s + 3i·8-s + (1 − 2i)10-s − 2·11-s + 2i·13-s − 2·14-s − 16-s + 2i·17-s − 19-s + (−2 − i)20-s − 2i·22-s + (3 + 4i)25-s − 2·26-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.5·4-s + (−0.894 − 0.447i)5-s + 0.755i·7-s + 1.06i·8-s + (0.316 − 0.632i)10-s − 0.603·11-s + 0.554i·13-s − 0.534·14-s − 0.250·16-s + 0.485i·17-s − 0.229·19-s + (−0.447 − 0.223i)20-s − 0.426i·22-s + (0.600 + 0.800i)25-s − 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 855,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248448 + 1.05244i\)
\(L(\frac12)\) \(\approx\) \(0.248448 + 1.05244i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
19 \( 1 + T \)
good2 \( 1 - iT - 2T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 18iT - 97T^{2} \)
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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

−10.79825486131797391212962993649, −9.422755247394498177531821990707, −8.619296314052961747359287112903, −7.905911542458527280977634980468, −7.24480507230017162053532898912, −6.20532620623201522542730246763, −5.41736537745405259602797701921, −4.45431662823273891089033482771, −3.13914195681282117764075202651, −1.87447109145379178149974946641, 0.49308327442812810118920035591, 2.19232002144848551942770727194, 3.32898224247665400544275299909, 3.96246734792274788932724653669, 5.28509586863728655778752439984, 6.58309485197631130609406322773, 7.36596128272417927145266405831, 7.85720717402839161488270301940, 9.128616988427608653322791127689, 10.28861374002305199843058850931

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