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Properties

Label 2-152-19.9-c1-0-3
Degree $2$
Conductor $152$
Sign $0.929 + 0.368i$
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  + (1.80 − 1.51i)3-s + (2.20 + 0.801i)5-s + (−2.07 + 3.59i)7-s + (0.447 − 2.53i)9-s + (−2.68 − 4.64i)11-s + (−2.26 − 1.89i)13-s + (5.19 − 1.89i)15-s + (0.962 + 5.46i)17-s + (−2.01 − 3.86i)19-s + (1.70 + 9.66i)21-s + (−1.68 + 0.613i)23-s + (0.374 + 0.314i)25-s + (0.497 + 0.862i)27-s + (−0.0374 + 0.212i)29-s + (1.20 − 2.08i)31-s + ⋯
L(s)  = 1  + (1.04 − 0.876i)3-s + (0.984 + 0.358i)5-s + (−0.785 + 1.36i)7-s + (0.149 − 0.846i)9-s + (−0.809 − 1.40i)11-s + (−0.627 − 0.526i)13-s + (1.34 − 0.488i)15-s + (0.233 + 1.32i)17-s + (−0.461 − 0.887i)19-s + (0.371 + 2.10i)21-s + (−0.351 + 0.127i)23-s + (0.0749 + 0.0628i)25-s + (0.0958 + 0.165i)27-s + (−0.00695 + 0.0394i)29-s + (0.216 − 0.374i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.929 + 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49299 - 0.285212i\)
\(L(\frac12)\) \(\approx\) \(1.49299 - 0.285212i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.01 + 3.86i)T \)
good3 \( 1 + (-1.80 + 1.51i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (-2.20 - 0.801i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (2.07 - 3.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.68 + 4.64i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.26 + 1.89i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.962 - 5.46i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (1.68 - 0.613i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.0374 - 0.212i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.20 + 2.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.54T + 37T^{2} \)
41 \( 1 + (1.33 - 1.12i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.55 - 0.564i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.19 - 6.75i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-5.94 + 2.16i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.77 - 10.0i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (9.39 - 3.41i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.55 + 8.81i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-10.0 - 3.67i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-9.55 + 8.01i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (4.29 - 3.60i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.65 + 6.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.15 - 3.48i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (3.09 + 17.5i)T + (-91.1 + 33.1i)T^{2} \)
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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

−13.11565094132311520468384373958, −12.36822722617507994114708564624, −10.81192097641816551764206405645, −9.651819254226226609992268079161, −8.690726280215010925337530198834, −7.907821753105860023664558227771, −6.34009069149458781327354793728, −5.67736402503196482384987382436, −2.99604443740712320033709385261, −2.33129441049299183802323487833, 2.39999167962025643455153672871, 3.94262005525714087965368174040, 5.00304731368984656617985884044, 6.80876143431078029743948527529, 7.86710069408372570984823976408, 9.473801196193294829589495779373, 9.761632271921194082914403528190, 10.44093530717282027318138601064, 12.31588902102399647764295460452, 13.35140163702724160836698995498

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