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Properties

Label 2-152-152.75-c5-0-35
Degree $2$
Conductor $152$
Sign $0.329 - 0.944i$
Analytic cond. $24.3783$
Root an. cond. $4.93744$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.42 − 1.60i)2-s + 14.2i·3-s + (26.8 + 17.4i)4-s − 9.24i·5-s + (22.8 − 77.0i)6-s + 124. i·7-s + (−117. − 137. i)8-s + 41.0·9-s + (−14.8 + 50.1i)10-s + 365.·11-s + (−248. + 381. i)12-s + 366.·13-s + (200. − 674. i)14-s + 131.·15-s + (414. + 936. i)16-s + 1.51e3·17-s + ⋯
L(s)  = 1  + (−0.958 − 0.284i)2-s + 0.911i·3-s + (0.838 + 0.545i)4-s − 0.165i·5-s + (0.259 − 0.874i)6-s + 0.958i·7-s + (−0.648 − 0.761i)8-s + 0.168·9-s + (−0.0470 + 0.158i)10-s + 0.909·11-s + (−0.497 + 0.764i)12-s + 0.601·13-s + (0.272 − 0.919i)14-s + 0.150·15-s + (0.405 + 0.914i)16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(24.3783\)
Root analytic conductor: \(4.93744\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :5/2),\ 0.329 - 0.944i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.434649453\)
\(L(\frac12)\) \(\approx\) \(1.434649453\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.42 + 1.60i)T \)
19 \( 1 + (794. + 1.35e3i)T \)
good3 \( 1 - 14.2iT - 243T^{2} \)
5 \( 1 + 9.24iT - 3.12e3T^{2} \)
7 \( 1 - 124. iT - 1.68e4T^{2} \)
11 \( 1 - 365.T + 1.61e5T^{2} \)
13 \( 1 - 366.T + 3.71e5T^{2} \)
17 \( 1 - 1.51e3T + 1.41e6T^{2} \)
23 \( 1 + 2.26e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.18e3T + 2.05e7T^{2} \)
31 \( 1 + 7.70e3T + 2.86e7T^{2} \)
37 \( 1 - 2.00e3T + 6.93e7T^{2} \)
41 \( 1 - 1.08e4iT - 1.15e8T^{2} \)
43 \( 1 - 6.49e3T + 1.47e8T^{2} \)
47 \( 1 - 1.15e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.10e4T + 4.18e8T^{2} \)
59 \( 1 - 3.40e4iT - 7.14e8T^{2} \)
61 \( 1 - 5.44e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.08e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.89e3T + 1.80e9T^{2} \)
73 \( 1 + 7.93e4T + 2.07e9T^{2} \)
79 \( 1 + 1.89e4T + 3.07e9T^{2} \)
83 \( 1 + 7.73e4T + 3.93e9T^{2} \)
89 \( 1 - 9.06e4iT - 5.58e9T^{2} \)
97 \( 1 + 4.17e4iT - 8.58e9T^{2} \)
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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

−12.04975614925500812886671993902, −11.03099933433064753673264187059, −10.15634612886113560513554407196, −9.109948609545554015750229460259, −8.669888907291803534284167449867, −7.12450913001998943030747781023, −5.83513671915291633981472550361, −4.23380752851853657244700258715, −2.88344437160108093905188605558, −1.19240740523927788170244128749, 0.828925073999091597255064534991, 1.67978001082139981721083605059, 3.68898045862585090262360220926, 5.81119642042133783209451656645, 6.92091408427511359617273348841, 7.49438859867897805795583958391, 8.590375990163366108232552079457, 9.852448822352555020181813052619, 10.69452112418909956722621416045, 11.79687928561516999253692049737

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