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Properties

Label 2-670-67.9-c1-0-7
Degree $2$
Conductor $670$
Sign $0.789 + 0.614i$
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 + (−0.212 − 0.0623i)3-s + (0.415 + 0.909i)4-s + (−0.654 + 0.755i)5-s + (0.144 + 0.167i)6-s + (0.563 + 0.362i)7-s + (0.142 − 0.989i)8-s + (−2.48 − 1.59i)9-s + (0.959 − 0.281i)10-s + (1.30 − 1.50i)11-s + (−0.0315 − 0.219i)12-s + (0.412 + 2.86i)13-s + (−0.278 − 0.609i)14-s + (0.186 − 0.119i)15-s + (−0.654 + 0.755i)16-s + (2.07 − 4.53i)17-s + ⋯
L(s)  = 1  + (−0.594 − 0.382i)2-s + (−0.122 − 0.0360i)3-s + (0.207 + 0.454i)4-s + (−0.292 + 0.337i)5-s + (0.0591 + 0.0683i)6-s + (0.212 + 0.136i)7-s + (0.0503 − 0.349i)8-s + (−0.827 − 0.531i)9-s + (0.303 − 0.0890i)10-s + (0.394 − 0.454i)11-s + (−0.00909 − 0.0632i)12-s + (0.114 + 0.795i)13-s + (−0.0743 − 0.162i)14-s + (0.0480 − 0.0309i)15-s + (−0.163 + 0.188i)16-s + (0.502 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.789 + 0.614i$
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: \((2,\ 670,\ (\ :1/2),\ 0.789 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.947369 - 0.325230i\)
\(L(\frac12)\) \(\approx\) \(0.947369 - 0.325230i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.75 + 7.70i)T \)
good3 \( 1 + (0.212 + 0.0623i)T + (2.52 + 1.62i)T^{2} \)
7 \( 1 + (-0.563 - 0.362i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (-1.30 + 1.50i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-0.412 - 2.86i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-2.07 + 4.53i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-1.33 + 0.857i)T + (7.89 - 17.2i)T^{2} \)
23 \( 1 + (-3.46 - 1.01i)T + (19.3 + 12.4i)T^{2} \)
29 \( 1 - 7.56T + 29T^{2} \)
31 \( 1 + (0.310 - 2.15i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 - 6.52T + 37T^{2} \)
41 \( 1 + (-2.35 + 5.14i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (1.02 - 2.23i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + (-7.99 - 2.34i)T + (39.5 + 25.4i)T^{2} \)
53 \( 1 + (-4.31 - 9.45i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-1.38 + 9.64i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (4.29 + 4.96i)T + (-8.68 + 60.3i)T^{2} \)
71 \( 1 + (2.67 + 5.85i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-5.16 - 5.95i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (2.06 + 14.3i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-1.06 + 1.22i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (3.66 - 1.07i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−10.51703732452291631316341191274, −9.352330312540506319969260048826, −8.927554997967880241722909320098, −7.908892262286416961046367879275, −6.96497401734248133516598538454, −6.12618193986710135199480413508, −4.86056910072910815748520611298, −3.51974909719281016735069268015, −2.63552214273033370492553680872, −0.869570002743491016487641347736, 1.08898342873908248513160049629, 2.73775836596238190491861133197, 4.22660131135211746131284822280, 5.33170987292963072291731942598, 6.10851386061640769950254925722, 7.27501452115874815555759394004, 8.158933418066930791949118981044, 8.597742563992293581073322147525, 9.750367984852772340298827812107, 10.52125467615745499547210339642

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