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Properties

Label 2-152-19.16-c1-0-2
Degree $2$
Conductor $152$
Sign $0.0878 + 0.996i$
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  + (−2.63 + 0.958i)3-s + (0.465 − 2.64i)5-s + (−0.731 − 1.26i)7-s + (3.72 − 3.12i)9-s + (2.54 − 4.40i)11-s + (−3.68 − 1.34i)13-s + (1.30 + 7.40i)15-s + (0.0943 + 0.0791i)17-s + (−1.78 − 3.97i)19-s + (3.14 + 2.63i)21-s + (1.23 + 6.98i)23-s + (−2.06 − 0.751i)25-s + (−2.60 + 4.51i)27-s + (−6.46 + 5.42i)29-s + (−3.09 − 5.35i)31-s + ⋯
L(s)  = 1  + (−1.52 + 0.553i)3-s + (0.208 − 1.18i)5-s + (−0.276 − 0.478i)7-s + (1.24 − 1.04i)9-s + (0.767 − 1.32i)11-s + (−1.02 − 0.371i)13-s + (0.337 + 1.91i)15-s + (0.0228 + 0.0192i)17-s + (−0.409 − 0.912i)19-s + (0.685 + 0.575i)21-s + (0.256 + 1.45i)23-s + (−0.413 − 0.150i)25-s + (−0.501 + 0.869i)27-s + (−1.20 + 1.00i)29-s + (−0.555 − 0.961i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0878 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0878 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.0878 + 0.996i$
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: \((2,\ 152,\ (\ :1/2),\ 0.0878 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438410 - 0.401447i\)
\(L(\frac12)\) \(\approx\) \(0.438410 - 0.401447i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.78 + 3.97i)T \)
good3 \( 1 + (2.63 - 0.958i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (-0.465 + 2.64i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.731 + 1.26i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.54 + 4.40i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.68 + 1.34i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.0943 - 0.0791i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.23 - 6.98i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (6.46 - 5.42i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (3.09 + 5.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.32T + 37T^{2} \)
41 \( 1 + (-9.35 + 3.40i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.262 - 1.48i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.06 + 1.73i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.04 + 5.95i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-6.47 - 5.43i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.19 - 6.75i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.33 + 6.15i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.403 - 2.28i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (11.1 - 4.04i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-2.80 + 1.02i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-1.93 - 3.35i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.22 - 2.99i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (9.50 + 7.97i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.67947681419608012127611237682, −11.56914674718444034398306264681, −10.98345951970808636336084785826, −9.726712806954889553975376247877, −8.947895420119938791069185489803, −7.23607800559268177279185216286, −5.85404268973810052354216494865, −5.17811204377810388636434554274, −3.94942833504526954962951356559, −0.70689692895179432355080240479, 2.21447272415464024713506889897, 4.48511940529411490761136540619, 5.93055570057610692620912876899, 6.69722690992619220480277641773, 7.43332218277996589791854669027, 9.488296541785469701730804778469, 10.40629327098602949356850236371, 11.29967925150914751089363358254, 12.29767170354130645581015346101, 12.68004566020791572649725838156

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