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Properties

Label 2-152-19.16-c1-0-4
Degree $2$
Conductor $152$
Sign $0.816 + 0.577i$
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  + (1.83 − 0.667i)3-s + (0.302 − 1.71i)5-s + (−0.962 − 1.66i)7-s + (0.622 − 0.522i)9-s + (−2.27 + 3.94i)11-s + (2.57 + 0.938i)13-s + (−0.589 − 3.34i)15-s + (3.84 + 3.22i)17-s + (−4.18 − 1.21i)19-s + (−2.87 − 2.41i)21-s + (0.198 + 1.12i)23-s + (1.85 + 0.675i)25-s + (−2.13 + 3.69i)27-s + (3.83 − 3.21i)29-s + (−0.999 − 1.73i)31-s + ⋯
L(s)  = 1  + (1.05 − 0.385i)3-s + (0.135 − 0.766i)5-s + (−0.363 − 0.630i)7-s + (0.207 − 0.174i)9-s + (−0.686 + 1.18i)11-s + (0.715 + 0.260i)13-s + (−0.152 − 0.863i)15-s + (0.931 + 0.781i)17-s + (−0.960 − 0.277i)19-s + (−0.628 − 0.527i)21-s + (0.0412 + 0.234i)23-s + (0.371 + 0.135i)25-s + (−0.411 + 0.711i)27-s + (0.712 − 0.597i)29-s + (−0.179 − 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38079 - 0.438704i\)
\(L(\frac12)\) \(\approx\) \(1.38079 - 0.438704i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.18 + 1.21i)T \)
good3 \( 1 + (-1.83 + 0.667i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (-0.302 + 1.71i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.962 + 1.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.27 - 3.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.57 - 0.938i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.84 - 3.22i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.198 - 1.12i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.83 + 3.21i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.999 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.06T + 37T^{2} \)
41 \( 1 + (10.3 - 3.77i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.05 + 6.01i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (6.72 - 5.63i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.00933 - 0.0529i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-2.83 - 2.37i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.60 + 14.7i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-9.72 + 8.15i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.359 - 2.03i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-0.670 + 0.243i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-10.5 + 3.85i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (4.99 + 8.64i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-15.2 - 5.54i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-0.285 - 0.239i)T + (16.8 + 95.5i)T^{2} \)
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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

−13.06184229492586150167349786239, −12.29637194667946387655154521489, −10.67484358648034656973614510506, −9.709415707179761448533835798378, −8.598248358075397460661257373766, −7.86669983248812612542346067775, −6.67125626422119666113563879526, −4.99270505521583404065667094245, −3.56628656621423052426002200406, −1.86160714341743249053584247660, 2.76649636234264983119842624828, 3.44968831373221540804239051968, 5.48444677741112176514942906128, 6.66771531935260861143674331503, 8.248551780341537680520730536486, 8.775430749801955493346016729582, 10.05180895161008177705289479245, 10.84470033069840042647841841979, 12.12844246164224617709784931540, 13.37933511701453274696912403195

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