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Properties

Label 2-155-155.4-c1-0-12
Degree $2$
Conductor $155$
Sign $0.688 + 0.725i$
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  + (2.53 − 0.823i)2-s + (−0.851 − 0.276i)3-s + (4.12 − 2.99i)4-s + (−1.95 + 1.07i)5-s − 2.38·6-s + (1.09 + 1.50i)7-s + (4.84 − 6.67i)8-s + (−1.77 − 1.29i)9-s + (−4.07 + 4.34i)10-s + (−2.65 + 1.93i)11-s + (−4.33 + 1.40i)12-s + (3.28 + 1.06i)13-s + (4.00 + 2.90i)14-s + (1.96 − 0.377i)15-s + (3.63 − 11.1i)16-s + (−3.91 + 5.38i)17-s + ⋯
L(s)  = 1  + (1.79 − 0.582i)2-s + (−0.491 − 0.159i)3-s + (2.06 − 1.49i)4-s + (−0.875 + 0.482i)5-s − 0.973·6-s + (0.412 + 0.568i)7-s + (1.71 − 2.35i)8-s + (−0.592 − 0.430i)9-s + (−1.28 + 1.37i)10-s + (−0.801 + 0.582i)11-s + (−1.25 + 0.406i)12-s + (0.911 + 0.296i)13-s + (1.07 + 0.777i)14-s + (0.507 − 0.0973i)15-s + (0.909 − 2.79i)16-s + (−0.949 + 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.688 + 0.725i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ 0.688 + 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04222 - 0.877360i\)
\(L(\frac12)\) \(\approx\) \(2.04222 - 0.877360i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.95 - 1.07i)T \)
31 \( 1 + (-3.17 - 4.57i)T \)
good2 \( 1 + (-2.53 + 0.823i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.851 + 0.276i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-1.09 - 1.50i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (2.65 - 1.93i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.28 - 1.06i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.91 - 5.38i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.38 + 4.27i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.973 + 1.33i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.535 + 1.64i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 7.99iT - 37T^{2} \)
41 \( 1 + (2.13 + 6.55i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (2.37 - 0.771i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-3.56 - 1.15i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.0789 + 0.108i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.897 + 2.76i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 - 2.09iT - 67T^{2} \)
71 \( 1 + (-11.9 - 8.68i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.13 + 2.93i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-5.70 - 4.14i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.71 + 0.881i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-0.0987 + 0.0717i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-6.84 - 9.42i)T + (-29.9 + 92.2i)T^{2} \)
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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

−12.62435142045499862379949209146, −12.03293668138857226576070293776, −10.99940549571231837687755327914, −10.78490931345319397302476664640, −8.587557435029426617433284793681, −6.92660871725265609613255722363, −6.07600406112074730115932473847, −4.89742245529081497405658506835, −3.77544688942121934388308015037, −2.38642328761542781831444751391, 3.10832024914889830828499970889, 4.42298239531668642000231384968, 5.18474190588063126560307666522, 6.27190246033619373397887771174, 7.63902371079453916225863553357, 8.338248734270959074460099438898, 10.84326314488570342136535922987, 11.34672911850856102729912386131, 12.18295816474401589593678357983, 13.46213206833532654581785935369

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