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Properties

Label 2-670-67.40-c1-0-10
Degree $2$
Conductor $670$
Sign $0.756 + 0.653i$
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  + (−0.654 − 0.755i)2-s + (0.200 − 0.439i)3-s + (−0.142 + 0.989i)4-s + (0.959 + 0.281i)5-s + (−0.463 + 0.136i)6-s + (−1.81 − 2.09i)7-s + (0.841 − 0.540i)8-s + (1.81 + 2.09i)9-s + (−0.415 − 0.909i)10-s + (2.88 + 0.847i)11-s + (0.406 + 0.261i)12-s + (2.41 + 1.54i)13-s + (−0.393 + 2.73i)14-s + (0.316 − 0.365i)15-s + (−0.959 − 0.281i)16-s + (−0.0159 − 0.111i)17-s + ⋯
L(s)  = 1  + (−0.463 − 0.534i)2-s + (0.115 − 0.253i)3-s + (−0.0711 + 0.494i)4-s + (0.429 + 0.125i)5-s + (−0.189 + 0.0555i)6-s + (−0.685 − 0.790i)7-s + (0.297 − 0.191i)8-s + (0.603 + 0.696i)9-s + (−0.131 − 0.287i)10-s + (0.870 + 0.255i)11-s + (0.117 + 0.0753i)12-s + (0.668 + 0.429i)13-s + (−0.105 + 0.732i)14-s + (0.0816 − 0.0942i)15-s + (−0.239 − 0.0704i)16-s + (−0.00387 − 0.0269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.756 + 0.653i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.756 + 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28903 - 0.479487i\)
\(L(\frac12)\) \(\approx\) \(1.28903 - 0.479487i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 + 0.755i)T \)
5 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (7.99 + 1.75i)T \)
good3 \( 1 + (-0.200 + 0.439i)T + (-1.96 - 2.26i)T^{2} \)
7 \( 1 + (1.81 + 2.09i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (-2.88 - 0.847i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-2.41 - 1.54i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (0.0159 + 0.111i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (2.93 - 3.38i)T + (-2.70 - 18.8i)T^{2} \)
23 \( 1 + (-2.21 + 4.85i)T + (-15.0 - 17.3i)T^{2} \)
29 \( 1 - 2.67T + 29T^{2} \)
31 \( 1 + (-5.13 + 3.30i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 - 3.50T + 37T^{2} \)
41 \( 1 + (-0.446 - 3.10i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (0.448 + 3.11i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (-1.08 + 2.37i)T + (-30.7 - 35.5i)T^{2} \)
53 \( 1 + (0.519 - 3.61i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-1.04 + 0.669i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-3.72 + 1.09i)T + (51.3 - 32.9i)T^{2} \)
71 \( 1 + (-0.510 + 3.54i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-13.2 + 3.88i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-3.81 - 2.44i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (0.483 + 0.141i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (4.76 + 10.4i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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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

−10.33165792841000364772807398183, −9.746786066232230196658327101700, −8.787134023116727587197003304964, −7.891027743612434687596005219354, −6.83340251840794100835912491428, −6.31328646557960545693566793023, −4.56700338013044041688098810258, −3.75667308140023456351235641921, −2.36012672856867949825291104569, −1.16783037057040043150875336146, 1.17049629920449784788962672072, 2.89897676940726264867693797174, 4.11827769913387317247407292014, 5.40310976021168818600666834584, 6.35390090864171060595553130403, 6.80476509414649616475971138743, 8.221766848134387952172538865117, 9.085632481712201594758299623774, 9.426171527936154489270080216909, 10.33008871254246030963283240508

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