Properties

Label 2-670-67.37-c1-0-13
Degree $2$
Conductor $670$
Sign $0.361 - 0.932i$
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  + (0.5 + 0.866i)2-s + 3.24·3-s + (−0.499 + 0.866i)4-s + 5-s + (1.62 + 2.81i)6-s + (−1.12 + 1.94i)7-s − 0.999·8-s + 7.55·9-s + (0.5 + 0.866i)10-s + (−2 + 3.46i)11-s + (−1.62 + 2.81i)12-s + (0.470 + 0.815i)13-s − 2.24·14-s + 3.24·15-s + (−0.5 − 0.866i)16-s + (−2.65 − 4.59i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + 1.87·3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (0.663 + 1.14i)6-s + (−0.425 + 0.736i)7-s − 0.353·8-s + 2.51·9-s + (0.158 + 0.273i)10-s + (−0.603 + 1.04i)11-s + (−0.468 + 0.812i)12-s + (0.130 + 0.226i)13-s − 0.601·14-s + 0.838·15-s + (−0.125 − 0.216i)16-s + (−0.643 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.361 - 0.932i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.361 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.64483 + 1.81141i\)
\(L(\frac12)\) \(\approx\) \(2.64483 + 1.81141i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 - T \)
67 \( 1 + (7.28 + 3.73i)T \)
good3 \( 1 - 3.24T + 3T^{2} \)
7 \( 1 + (1.12 - 1.94i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.470 - 0.815i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.65 + 4.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.30 + 7.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.124 + 0.215i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0586 + 0.101i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.09 + 7.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.21 - 5.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.27 - 3.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-6.12 + 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.175T + 53T^{2} \)
59 \( 1 - 6.49T + 59T^{2} \)
61 \( 1 + (-5.15 - 8.92i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-2.34 + 4.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.90 + 5.02i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.412 + 0.713i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.12 - 8.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + (6.55 + 11.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.21543464406560023811757466077, −9.416472248195583654915408370315, −8.951475637610369013959891923982, −8.148445836386590807512788937646, −7.15689618450410337398788721895, −6.56214875070945132132297286356, −4.97706417389587619339615081198, −4.20858938532067516266792255551, −2.68261075957153917046498885239, −2.38600852777455654129628756125, 1.54442845858244255029643721241, 2.64916966730529240752113311068, 3.59549658250935952564045527986, 4.19775664039105472473519965056, 5.82264551367987752027364724860, 6.88836598115259082066625229847, 8.201491090440185241578771532332, 8.479760086723009320990965176949, 9.551809864393625363649489717825, 10.40652448463416181453914935526

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