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Properties

Label 2-152-19.9-c1-0-4
Degree $2$
Conductor $152$
Sign $0.714 + 0.700i$
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  + (1.67 − 1.40i)3-s + (−0.826 − 0.300i)5-s + (0.826 − 1.43i)7-s + (0.307 − 1.74i)9-s + (1.76 + 3.05i)11-s + (−3.78 − 3.17i)13-s + (−1.80 + 0.657i)15-s + (0.450 + 2.55i)17-s + (0.354 + 4.34i)19-s + (−0.627 − 3.55i)21-s + (4.53 − 1.64i)23-s + (−3.23 − 2.71i)25-s + (1.34 + 2.32i)27-s + (−1.19 + 6.77i)29-s + (−1.02 + 1.76i)31-s + ⋯
L(s)  = 1  + (0.966 − 0.810i)3-s + (−0.369 − 0.134i)5-s + (0.312 − 0.540i)7-s + (0.102 − 0.582i)9-s + (0.532 + 0.922i)11-s + (−1.05 − 0.881i)13-s + (−0.466 + 0.169i)15-s + (0.109 + 0.620i)17-s + (0.0813 + 0.996i)19-s + (−0.136 − 0.775i)21-s + (0.945 − 0.343i)23-s + (−0.647 − 0.543i)25-s + (0.257 + 0.446i)27-s + (−0.221 + 1.25i)29-s + (−0.183 + 0.317i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.714 + 0.700i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.714 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30063 - 0.531156i\)
\(L(\frac12)\) \(\approx\) \(1.30063 - 0.531156i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-0.354 - 4.34i)T \)
good3 \( 1 + (-1.67 + 1.40i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (0.826 + 0.300i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.826 + 1.43i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.76 - 3.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.78 + 3.17i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.450 - 2.55i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-4.53 + 1.64i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.19 - 6.77i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.02 - 1.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.18T + 37T^{2} \)
41 \( 1 + (-1.33 + 1.11i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (7.50 + 2.73i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.507 + 2.87i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (11.9 - 4.33i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.31 + 7.46i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-10.4 + 3.80i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.79 + 15.8i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-12.7 - 4.63i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-3.63 + 3.04i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (0.933 - 0.783i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.93 + 5.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.88 + 6.61i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.98 + 11.2i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.61872229631788494715879415202, −12.41425681598860186993352374766, −10.79453387662249200427696351704, −9.726396306235200950620635961932, −8.426100870583182581004256974444, −7.67159155317434904593978504481, −6.86760311265819032156847237331, −4.97371770631896980555020523649, −3.45479081988621444808383464768, −1.79194421800063348002325011051, 2.64611949732025841826463406682, 3.89247652386135324755501241241, 5.15016315537300656932509088054, 6.86506098787827665692427074576, 8.148551383238913141846803829031, 9.157165109705557708302298284241, 9.671075701169648580247481630675, 11.27957664462538265236282925193, 11.82027284710865720927212271319, 13.40138412980715007434581470831

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