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Properties

Label 2-670-67.9-c1-0-17
Degree $2$
Conductor $670$
Sign $0.333 + 0.942i$
Analytic cond. $5.34997$
Root an. cond. $2.31300$
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  + (0.841 + 0.540i)2-s + (−0.673 − 0.197i)3-s + (0.415 + 0.909i)4-s + (0.654 − 0.755i)5-s + (−0.459 − 0.530i)6-s + (−2.18 − 1.40i)7-s + (−0.142 + 0.989i)8-s + (−2.10 − 1.35i)9-s + (0.959 − 0.281i)10-s + (0.515 − 0.595i)11-s + (−0.0998 − 0.694i)12-s + (−0.545 − 3.79i)13-s + (−1.08 − 2.36i)14-s + (−0.590 + 0.379i)15-s + (−0.654 + 0.755i)16-s + (1.85 − 4.05i)17-s + ⋯
L(s)  = 1  + (0.594 + 0.382i)2-s + (−0.388 − 0.114i)3-s + (0.207 + 0.454i)4-s + (0.292 − 0.337i)5-s + (−0.187 − 0.216i)6-s + (−0.826 − 0.531i)7-s + (−0.0503 + 0.349i)8-s + (−0.703 − 0.451i)9-s + (0.303 − 0.0890i)10-s + (0.155 − 0.179i)11-s + (−0.0288 − 0.200i)12-s + (−0.151 − 1.05i)13-s + (−0.288 − 0.632i)14-s + (−0.152 + 0.0979i)15-s + (−0.163 + 0.188i)16-s + (0.449 − 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.333 + 0.942i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14432 - 0.808792i\)
\(L(\frac12)\) \(\approx\) \(1.14432 - 0.808792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (-6.47 + 5.00i)T \)
good3 \( 1 + (0.673 + 0.197i)T + (2.52 + 1.62i)T^{2} \)
7 \( 1 + (2.18 + 1.40i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (-0.515 + 0.595i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (0.545 + 3.79i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-1.85 + 4.05i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-5.31 + 3.41i)T + (7.89 - 17.2i)T^{2} \)
23 \( 1 + (0.758 + 0.222i)T + (19.3 + 12.4i)T^{2} \)
29 \( 1 + 4.08T + 29T^{2} \)
31 \( 1 + (0.771 - 5.36i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + 8.64T + 37T^{2} \)
41 \( 1 + (-0.674 + 1.47i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-4.09 + 8.95i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + (-4.76 - 1.39i)T + (39.5 + 25.4i)T^{2} \)
53 \( 1 + (0.442 + 0.968i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-0.178 + 1.24i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-7.22 - 8.34i)T + (-8.68 + 60.3i)T^{2} \)
71 \( 1 + (-4.57 - 10.0i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (2.57 + 2.97i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (0.888 + 6.18i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (2.30 - 2.66i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (6.15 - 1.80i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−10.36789920961399544172223273751, −9.445597402248268124663837226120, −8.622481067512128861555311686551, −7.35695401832782464793349333640, −6.79404511845244715519474129506, −5.56566552793485794469470143827, −5.26547306056932469757545159632, −3.66064895791673240230660782910, −2.88262459948124846490770206739, −0.63407107128261886056046098674, 1.87447587936192811695540937994, 3.05001999166635983153732939728, 4.06912223168349351693951019762, 5.43981156365869990870078436410, 5.93532482977553343163661174430, 6.84706391643697730906682902300, 8.040983074227203501395966930472, 9.318049244519473540086952127046, 9.850192201449333015544925093929, 10.82261858803389008070930478766

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