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Properties

Label 2-152-152.75-c1-0-5
Degree $2$
Conductor $152$
Sign $0.912 - 0.408i$
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  + (−1.17 + 0.792i)2-s − 1.26i·3-s + (0.743 − 1.85i)4-s + 3.51i·5-s + (1 + 1.47i)6-s − 2.23i·7-s + (0.601 + 2.76i)8-s + 1.40·9-s + (−2.78 − 4.12i)10-s + 2.89·11-s + (−2.34 − 0.937i)12-s + 6.30·13-s + (1.77 + 2.61i)14-s + 4.43·15-s + (−2.89 − 2.76i)16-s − 4.79·17-s + ⋯
L(s)  = 1  + (−0.828 + 0.560i)2-s − 0.728i·3-s + (0.371 − 0.928i)4-s + 1.57i·5-s + (0.408 + 0.603i)6-s − 0.845i·7-s + (0.212 + 0.977i)8-s + 0.469·9-s + (−0.882 − 1.30i)10-s + 0.872·11-s + (−0.676 − 0.270i)12-s + 1.74·13-s + (0.473 + 0.699i)14-s + 1.14·15-s + (−0.723 − 0.690i)16-s − 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.912 - 0.408i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.912 - 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.832185 + 0.177838i\)
\(L(\frac12)\) \(\approx\) \(0.832185 + 0.177838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 - 0.792i)T \)
19 \( 1 + (-0.895 - 4.26i)T \)
good3 \( 1 + 1.26iT - 3T^{2} \)
5 \( 1 - 3.51iT - 5T^{2} \)
7 \( 1 + 2.23iT - 7T^{2} \)
11 \( 1 - 2.89T + 11T^{2} \)
13 \( 1 - 6.30T + 13T^{2} \)
17 \( 1 + 4.79T + 17T^{2} \)
23 \( 1 + 0.524iT - 23T^{2} \)
29 \( 1 + 0.415T + 29T^{2} \)
31 \( 1 + 1.20T + 31T^{2} \)
37 \( 1 + 5.88T + 37T^{2} \)
41 \( 1 + 4.87iT - 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 4.27iT - 47T^{2} \)
53 \( 1 + 6.05T + 53T^{2} \)
59 \( 1 - 8.08iT - 59T^{2} \)
61 \( 1 - 8.38iT - 61T^{2} \)
67 \( 1 + 9.79iT - 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 7.76T + 73T^{2} \)
79 \( 1 - 7.33T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 12.1iT - 89T^{2} \)
97 \( 1 - 2.19iT - 97T^{2} \)
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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

−13.50538529979423743216202916454, −11.71077299296906940886551198616, −10.78280497035339202325248284827, −10.20193334225964975960730927305, −8.749947164833994760098899168652, −7.53699034863407855576274257615, −6.74270851361341054006055784802, −6.24460713044118081077069465629, −3.77905431970872009933633846973, −1.64962453199860879846504505147, 1.48996415324906174572912104417, 3.75099645536233009548356757254, 4.86058757100017245891316146689, 6.53773729613293747511879699525, 8.382998536360891458685669663192, 8.979111253069402394513259396324, 9.503679705357900724163168498389, 10.94725584319405234744814259206, 11.73203713809189707470464920181, 12.79391017318517494604320062928

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