Properties

Label 2-155-155.19-c1-0-2
Degree $2$
Conductor $155$
Sign $-0.741 - 0.671i$
Analytic cond. $1.23768$
Root an. cond. $1.11251$
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  + (−0.203 + 0.280i)2-s + (−1.27 + 2.87i)3-s + (0.580 + 1.78i)4-s + (2.15 + 0.608i)5-s + (−0.544 − 0.943i)6-s + (−2.13 − 1.91i)7-s + (−1.27 − 0.415i)8-s + (−4.60 − 5.10i)9-s + (−0.609 + 0.479i)10-s + (1.68 + 0.358i)11-s + (−5.87 − 0.617i)12-s + (5.44 − 0.571i)13-s + (0.972 − 0.206i)14-s + (−4.49 + 5.39i)15-s + (−2.66 + 1.93i)16-s + (0.244 + 1.15i)17-s + ⋯
L(s)  = 1  + (−0.144 + 0.198i)2-s + (−0.737 + 1.65i)3-s + (0.290 + 0.893i)4-s + (0.962 + 0.272i)5-s + (−0.222 − 0.385i)6-s + (−0.805 − 0.725i)7-s + (−0.452 − 0.146i)8-s + (−1.53 − 1.70i)9-s + (−0.192 + 0.151i)10-s + (0.508 + 0.108i)11-s + (−1.69 − 0.178i)12-s + (1.50 − 0.158i)13-s + (0.259 − 0.0552i)14-s + (−1.16 + 1.39i)15-s + (−0.666 + 0.483i)16-s + (0.0594 + 0.279i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.741 - 0.671i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ -0.741 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337679 + 0.876162i\)
\(L(\frac12)\) \(\approx\) \(0.337679 + 0.876162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.15 - 0.608i)T \)
31 \( 1 + (-5.43 + 1.21i)T \)
good2 \( 1 + (0.203 - 0.280i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (1.27 - 2.87i)T + (-2.00 - 2.22i)T^{2} \)
7 \( 1 + (2.13 + 1.91i)T + (0.731 + 6.96i)T^{2} \)
11 \( 1 + (-1.68 - 0.358i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-5.44 + 0.571i)T + (12.7 - 2.70i)T^{2} \)
17 \( 1 + (-0.244 - 1.15i)T + (-15.5 + 6.91i)T^{2} \)
19 \( 1 + (0.251 - 2.39i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (6.26 + 2.03i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.45 - 1.06i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (-0.390 + 0.225i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.11 - 1.83i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (-3.70 - 0.389i)T + (42.0 + 8.94i)T^{2} \)
47 \( 1 + (-2.94 - 4.05i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.06 + 1.86i)T + (5.54 - 52.7i)T^{2} \)
59 \( 1 + (12.4 + 5.55i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 - 8.17T + 61T^{2} \)
67 \( 1 + (-8.49 - 4.90i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.09 + 2.33i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (-0.448 + 2.11i)T + (-66.6 - 29.6i)T^{2} \)
79 \( 1 + (-9.59 + 2.04i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (2.37 + 5.33i)T + (-55.5 + 61.6i)T^{2} \)
89 \( 1 + (5.21 + 16.0i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.66 - 1.84i)T + (78.4 - 57.0i)T^{2} \)
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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

−13.36241729477154743154798150790, −12.15714386998780038201073694250, −11.04181030424042447395865125920, −10.28373905399837846016391247996, −9.523024207005962477001119836724, −8.435769117051605036189942568044, −6.52047292632219825027838376715, −5.98144891451591349554660999722, −4.17450492034715461880428146120, −3.35029087752451882908204799272, 1.15706793654418470935796642845, 2.38164081974799599582861671688, 5.52012645524742205942978649628, 6.18741334789453263461063697863, 6.69388289736618356350974720615, 8.487609740114407932899904846277, 9.515821463203165570848731416818, 10.75358521270671830006144823133, 11.74089348392151393879079742298, 12.46236377049040801534036574585

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