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Properties

Label 2-152-152.27-c1-0-0
Degree $2$
Conductor $152$
Sign $0.0950 - 0.995i$
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  + (−0.833 − 1.14i)2-s + (−2.03 − 1.17i)3-s + (−0.611 + 1.90i)4-s + (−1.50 − 0.869i)5-s + (0.352 + 3.29i)6-s + 2.63i·7-s + (2.68 − 0.888i)8-s + (1.24 + 2.16i)9-s + (0.261 + 2.44i)10-s − 3.51·11-s + (3.47 − 3.15i)12-s + (3.13 + 5.42i)13-s + (3.00 − 2.19i)14-s + (2.03 + 3.53i)15-s + (−3.25 − 2.32i)16-s + (−0.535 + 0.926i)17-s + ⋯
L(s)  = 1  + (−0.589 − 0.807i)2-s + (−1.17 − 0.676i)3-s + (−0.305 + 0.952i)4-s + (−0.673 − 0.388i)5-s + (0.143 + 1.34i)6-s + 0.995i·7-s + (0.949 − 0.314i)8-s + (0.416 + 0.721i)9-s + (0.0826 + 0.773i)10-s − 1.05·11-s + (1.00 − 0.909i)12-s + (0.868 + 1.50i)13-s + (0.804 − 0.586i)14-s + (0.526 + 0.911i)15-s + (−0.813 − 0.582i)16-s + (−0.129 + 0.224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0950 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0950 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.0950 - 0.995i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.0950 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0926681 + 0.0842420i\)
\(L(\frac12)\) \(\approx\) \(0.0926681 + 0.0842420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.833 + 1.14i)T \)
19 \( 1 + (3.79 + 2.13i)T \)
good3 \( 1 + (2.03 + 1.17i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.50 + 0.869i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.63iT - 7T^{2} \)
11 \( 1 + 3.51T + 11T^{2} \)
13 \( 1 + (-3.13 - 5.42i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.535 - 0.926i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (5.79 - 3.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.579 + 1.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.82T + 31T^{2} \)
37 \( 1 + 3.55T + 37T^{2} \)
41 \( 1 + (4.54 + 2.62i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.164 - 0.284i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.68 + 3.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.526 + 0.912i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.45 - 4.87i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.2 - 6.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.3 + 7.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.554 - 0.960i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.98 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.02 + 1.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 + (-2.18 + 1.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.354 - 0.204i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.72805885554446968435699312733, −11.95987301736833302748761330533, −11.49887498769313826843991068235, −10.53943814854396236286601105950, −9.041787460549424780886413120331, −8.219515295622742431846791662346, −6.93161320666219283888179486481, −5.61929825862714975101117388239, −4.13126026038661028781607612556, −1.99257105981563773042164781930, 0.16505910892259694858326558938, 3.97420619287004106408440646164, 5.25361864328097425468328895088, 6.20886336866102181747099704962, 7.51738016509413175734786263825, 8.306456055358528287731256505773, 10.16185615963089695784705509835, 10.55132708831940911827402620218, 11.19168242070023764490071158135, 12.77433713190644821652020534972

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