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Properties

Label 2-152-152.125-c1-0-16
Degree $2$
Conductor $152$
Sign $-0.787 + 0.616i$
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  + (0.684 + 1.23i)2-s + (−1.95 − 1.13i)3-s + (−1.06 + 1.69i)4-s + (−3.00 − 1.73i)5-s + (0.0593 − 3.19i)6-s − 2.66·7-s + (−2.82 − 0.157i)8-s + (1.05 + 1.83i)9-s + (0.0911 − 4.90i)10-s + 5.21i·11-s + (3.99 − 2.11i)12-s + (3.87 − 2.23i)13-s + (−1.82 − 3.29i)14-s + (3.92 + 6.79i)15-s + (−1.73 − 3.60i)16-s + (−0.0984 + 0.170i)17-s + ⋯
L(s)  = 1  + (0.483 + 0.875i)2-s + (−1.13 − 0.653i)3-s + (−0.531 + 0.846i)4-s + (−1.34 − 0.776i)5-s + (0.0242 − 1.30i)6-s − 1.00·7-s + (−0.998 − 0.0556i)8-s + (0.353 + 0.611i)9-s + (0.0288 − 1.55i)10-s + 1.57i·11-s + (1.15 − 0.610i)12-s + (1.07 − 0.620i)13-s + (−0.487 − 0.881i)14-s + (1.01 + 1.75i)15-s + (−0.434 − 0.900i)16-s + (−0.0238 + 0.0413i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.787 + 0.616i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ -0.787 + 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0145547 - 0.0422054i\)
\(L(\frac12)\) \(\approx\) \(0.0145547 - 0.0422054i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.684 - 1.23i)T \)
19 \( 1 + (3.73 + 2.24i)T \)
good3 \( 1 + (1.95 + 1.13i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.00 + 1.73i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.66T + 7T^{2} \)
11 \( 1 - 5.21iT - 11T^{2} \)
13 \( 1 + (-3.87 + 2.23i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.0984 - 0.170i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.60 + 4.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.51 - 2.60i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.47T + 31T^{2} \)
37 \( 1 + 4.11iT - 37T^{2} \)
41 \( 1 + (1.67 - 2.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.11 - 0.644i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.511 + 0.885i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.19 - 4.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.43 + 0.828i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.70 + 0.983i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.10 + 5.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.35 - 5.80i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.33 + 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.342 - 0.592i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.58iT - 83T^{2} \)
89 \( 1 + (0.509 + 0.881i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.90 - 15.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.70287055785343502007393749868, −12.11027668527919784724769005710, −10.93692231846388814309326393761, −9.215308663673809056112427041720, −8.029576988597245487357508690457, −7.04343384847689708202648004613, −6.21378611072170211178555745330, −4.91545111771195235377415952544, −3.81004071191314574906753950279, −0.04170656060569691935395276149, 3.43477164628109099729157420338, 3.99483637242765034459521153936, 5.73481750336188017845051724271, 6.48506024482957367924202536949, 8.398892945259645305127763186038, 9.784194399695877068278545131585, 10.98765240870924830477717171296, 11.15259892632920380487205577465, 11.97780984310105298019145303517, 13.18966769266577795660047922140

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