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Properties

Label 2-160-5.4-c5-0-7
Degree $2$
Conductor $160$
Sign $0.564 - 0.825i$
Analytic cond. $25.6614$
Root an. cond. $5.06570$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.01i·3-s + (−46.1 − 31.5i)5-s − 15.3i·7-s + 217.·9-s − 576.·11-s + 607. i·13-s + (158. − 231. i)15-s − 2.01e3i·17-s + 2.42e3·19-s + 77.2·21-s + 4.35e3i·23-s + (1.13e3 + 2.91e3i)25-s + 2.31e3i·27-s − 1.22e3·29-s + 7.81e3·31-s + ⋯
L(s)  = 1  + 0.321i·3-s + (−0.825 − 0.564i)5-s − 0.118i·7-s + 0.896·9-s − 1.43·11-s + 0.996i·13-s + (0.181 − 0.265i)15-s − 1.68i·17-s + 1.53·19-s + 0.0382·21-s + 1.71i·23-s + (0.362 + 0.931i)25-s + 0.610i·27-s − 0.270·29-s + 1.46·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.564 - 0.825i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.419420126\)
\(L(\frac12)\) \(\approx\) \(1.419420126\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (46.1 + 31.5i)T \)
good3 \( 1 - 5.01iT - 243T^{2} \)
7 \( 1 + 15.3iT - 1.68e4T^{2} \)
11 \( 1 + 576.T + 1.61e5T^{2} \)
13 \( 1 - 607. iT - 3.71e5T^{2} \)
17 \( 1 + 2.01e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.42e3T + 2.47e6T^{2} \)
23 \( 1 - 4.35e3iT - 6.43e6T^{2} \)
29 \( 1 + 1.22e3T + 2.05e7T^{2} \)
31 \( 1 - 7.81e3T + 2.86e7T^{2} \)
37 \( 1 - 3.80e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.47e3T + 1.15e8T^{2} \)
43 \( 1 - 1.47e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.04e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.09e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.11e4T + 7.14e8T^{2} \)
61 \( 1 + 4.48e3T + 8.44e8T^{2} \)
67 \( 1 - 4.54e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.03e4T + 1.80e9T^{2} \)
73 \( 1 + 3.89e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.32e4T + 3.07e9T^{2} \)
83 \( 1 + 3.47e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.78e4T + 5.58e9T^{2} \)
97 \( 1 + 1.51e5iT - 8.58e9T^{2} \)
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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

−11.94354280436680932657606699174, −11.35073180982085460540659346705, −9.944452311795918793581637585318, −9.257176762335725461182920521968, −7.75840540232578771004749243351, −7.22636036195266730004358492473, −5.26340456054372085771413507420, −4.48467439362563412727757133594, −3.07157707416393067235942446584, −1.07361257774170848939292074699, 0.58149714460924359751501040320, 2.49738169284544327248127053745, 3.81227829084268295028356291385, 5.25315844059530392335169560997, 6.66057852700597048302285260202, 7.74074048775605787758484587160, 8.328724723437521400582918381473, 10.30017542262080165704147406215, 10.50827258374487108107650395690, 12.01629218305354608435101897609

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