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Properties

Label 2-152-152.107-c1-0-5
Degree $2$
Conductor $152$
Sign $-0.234 - 0.972i$
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  + (−0.853 + 1.12i)2-s + (0.179 − 0.103i)3-s + (−0.543 − 1.92i)4-s + (−0.520 + 0.300i)5-s + (−0.0362 + 0.290i)6-s + 4.27i·7-s + (2.63 + 1.03i)8-s + (−1.47 + 2.56i)9-s + (0.105 − 0.843i)10-s + 5.11·11-s + (−0.296 − 0.288i)12-s + (−1.55 + 2.68i)13-s + (−4.81 − 3.64i)14-s + (−0.0621 + 0.107i)15-s + (−3.40 + 2.09i)16-s + (−1.52 − 2.64i)17-s + ⋯
L(s)  = 1  + (−0.603 + 0.797i)2-s + (0.103 − 0.0597i)3-s + (−0.271 − 0.962i)4-s + (−0.232 + 0.134i)5-s + (−0.0148 + 0.118i)6-s + 1.61i·7-s + (0.931 + 0.364i)8-s + (−0.492 + 0.853i)9-s + (0.0333 − 0.266i)10-s + 1.54·11-s + (−0.0855 − 0.0833i)12-s + (−0.430 + 0.745i)13-s + (−1.28 − 0.974i)14-s + (−0.0160 + 0.0277i)15-s + (−0.852 + 0.522i)16-s + (−0.371 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.234 - 0.972i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ -0.234 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.493236 + 0.626574i\)
\(L(\frac12)\) \(\approx\) \(0.493236 + 0.626574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.853 - 1.12i)T \)
19 \( 1 + (-3.31 + 2.83i)T \)
good3 \( 1 + (-0.179 + 0.103i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.520 - 0.300i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.27iT - 7T^{2} \)
11 \( 1 - 5.11T + 11T^{2} \)
13 \( 1 + (1.55 - 2.68i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.52 + 2.64i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.65 + 2.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.77 + 6.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.59T + 31T^{2} \)
37 \( 1 + 4.44T + 37T^{2} \)
41 \( 1 + (0.253 - 0.146i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.892 + 1.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.79 - 3.34i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.16 + 8.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.849 - 0.490i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.08 + 2.93i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.98 - 3.45i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.173 + 0.300i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.61 - 7.99i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.63 + 6.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.38T + 83T^{2} \)
89 \( 1 + (-5.44 - 3.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.87 + 4.54i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.73965771011731046892950302190, −11.87299488342812997065926844443, −11.51752687732521682794743236169, −9.807267263044055705714664610591, −9.010133956552059374498666551103, −8.237306694931160895893795417030, −6.90810693659865467505129669890, −5.89811966856097757508747174761, −4.68180013159830301697552604305, −2.28009210298575760415317653561, 1.04620138751735297943000885760, 3.45989628214527783810025146515, 4.23033217635522778262679879189, 6.50842643238865047721729874852, 7.66125189113454172784180237282, 8.686222922713149719827967806075, 9.848691335006027782057501660552, 10.50511550178527227233143508340, 11.80556725531847569701716138067, 12.27852682198502118340397819843

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