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Properties

Label 2-154-11.5-c1-0-1
Degree $2$
Conductor $154$
Sign $-0.174 - 0.984i$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
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  + (0.309 + 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 2.48i)5-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.618 − 1.90i)9-s − 2.61·10-s + (3.23 − 0.726i)11-s − 0.999·12-s + (1.07 + 3.30i)13-s + (−0.809 − 0.587i)14-s + (−2.11 + 1.53i)15-s + (0.309 − 0.951i)16-s + (2.30 − 7.10i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.361 + 1.11i)5-s + (−0.126 + 0.388i)6-s + (−0.305 + 0.222i)7-s + (−0.286 − 0.207i)8-s + (−0.206 − 0.634i)9-s − 0.827·10-s + (0.975 − 0.219i)11-s − 0.288·12-s + (0.297 + 0.915i)13-s + (−0.216 − 0.157i)14-s + (−0.546 + 0.397i)15-s + (0.0772 − 0.237i)16-s + (0.560 − 1.72i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.174 - 0.984i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ -0.174 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.829646 + 0.989296i\)
\(L(\frac12)\) \(\approx\) \(0.829646 + 0.989296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-3.23 + 0.726i)T \)
good3 \( 1 + (-0.809 - 0.587i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.809 - 2.48i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1.07 - 3.30i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.30 + 7.10i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.42 - 3.94i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 8.23T + 23T^{2} \)
29 \( 1 + (-0.427 + 0.310i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.927 + 2.85i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.35 + 4.61i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2 - 1.45i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.38T + 43T^{2} \)
47 \( 1 + (-0.5 - 0.363i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.92 + 12.0i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.35 - 6.06i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.20 - 6.79i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 8.70T + 67T^{2} \)
71 \( 1 + (0.927 - 2.85i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.54 + 4.02i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.21 - 3.75i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.95 - 6.01i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 3.09T + 89T^{2} \)
97 \( 1 + (-1.09 - 3.35i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.84804496202539874205304776924, −11.97611851788576830296538958637, −11.58654336653336261625239292013, −9.810321577258027655152126214904, −9.255186003169956891201052641844, −7.86366761699286027855856586130, −6.82825064605553141589362189990, −5.91482284188335634080481322946, −4.01589175979856366600615350466, −3.12244196765715953694966201856, 1.42331276958318530634976813559, 3.36381543741828596272332928202, 4.62627814545694771610292239322, 5.97345499454014856589545887521, 7.79002480995481914238087977225, 8.527908452602367184441024177544, 9.627938819585908860523857094915, 10.73196167823892014200859307338, 11.97685175169992742698330689581, 12.65227778041184209381946720929

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