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Properties

Label 2-152-8.5-c1-0-9
Degree $2$
Conductor $152$
Sign $0.845 - 0.533i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
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  + (1.33 + 0.466i)2-s − 0.579i·3-s + (1.56 + 1.24i)4-s + 2.10i·5-s + (0.269 − 0.773i)6-s − 2.73·7-s + (1.50 + 2.39i)8-s + 2.66·9-s + (−0.983 + 2.81i)10-s − 4.66i·11-s + (0.720 − 0.906i)12-s − 4.47i·13-s + (−3.65 − 1.27i)14-s + 1.22·15-s + (0.900 + 3.89i)16-s − 6.85·17-s + ⋯
L(s)  = 1  + (0.944 + 0.329i)2-s − 0.334i·3-s + (0.782 + 0.622i)4-s + 0.943i·5-s + (0.110 − 0.315i)6-s − 1.03·7-s + (0.533 + 0.845i)8-s + 0.888·9-s + (−0.310 + 0.890i)10-s − 1.40i·11-s + (0.208 − 0.261i)12-s − 1.24i·13-s + (−0.975 − 0.340i)14-s + 0.315·15-s + (0.225 + 0.974i)16-s − 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.845 - 0.533i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.845 - 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70219 + 0.492288i\)
\(L(\frac12)\) \(\approx\) \(1.70219 + 0.492288i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 - 0.466i)T \)
19 \( 1 - iT \)
good3 \( 1 + 0.579iT - 3T^{2} \)
5 \( 1 - 2.10iT - 5T^{2} \)
7 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 + 4.66iT - 11T^{2} \)
13 \( 1 + 4.47iT - 13T^{2} \)
17 \( 1 + 6.85T + 17T^{2} \)
23 \( 1 + 1.20T + 23T^{2} \)
29 \( 1 - 9.57iT - 29T^{2} \)
31 \( 1 + 5.35T + 31T^{2} \)
37 \( 1 + 1.09iT - 37T^{2} \)
41 \( 1 - 7.33T + 41T^{2} \)
43 \( 1 + 7.64iT - 43T^{2} \)
47 \( 1 - 7.56T + 47T^{2} \)
53 \( 1 + 3.11iT - 53T^{2} \)
59 \( 1 - 10.2iT - 59T^{2} \)
61 \( 1 + 0.722iT - 61T^{2} \)
67 \( 1 - 6.13iT - 67T^{2} \)
71 \( 1 - 4.62T + 71T^{2} \)
73 \( 1 + 6.19T + 73T^{2} \)
79 \( 1 + 3.26T + 79T^{2} \)
83 \( 1 - 8.97iT - 83T^{2} \)
89 \( 1 - 0.620T + 89T^{2} \)
97 \( 1 + 1.67T + 97T^{2} \)
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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

−13.10296106919563050751073051958, −12.51855769886047068389177731425, −11.02399075334968334390684592819, −10.49625037531145473595243894834, −8.754565049937780458586214253989, −7.33034853861619079828541511975, −6.62265688521618533207421365538, −5.64985062163604890693457449521, −3.79601122281903407543512973027, −2.77437871939406430210849309083, 2.07194012362618841419490790404, 4.19873851804973751801403548906, 4.61664797569637005150615373509, 6.34151011513925337699953367823, 7.22344986821642834468081326907, 9.298317082890285861452051885819, 9.734262121853649247645371098621, 11.05768725311309436063408139848, 12.25249718533183788903043353689, 12.88281377544513619220192896670

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