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Properties

Label 2-670-67.25-c1-0-15
Degree $2$
Conductor $670$
Sign $0.889 + 0.457i$
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  + (0.142 + 0.989i)2-s + (1.98 − 2.29i)3-s + (−0.959 + 0.281i)4-s + (0.841 − 0.540i)5-s + (2.55 + 1.64i)6-s + (0.266 + 1.85i)7-s + (−0.415 − 0.909i)8-s + (−0.883 − 6.14i)9-s + (0.654 + 0.755i)10-s + (1.03 − 0.662i)11-s + (−1.26 + 2.76i)12-s + (0.0783 − 0.171i)13-s + (−1.79 + 0.526i)14-s + (0.431 − 3.00i)15-s + (0.841 − 0.540i)16-s + (5.49 + 1.61i)17-s + ⋯
L(s)  = 1  + (0.100 + 0.699i)2-s + (1.14 − 1.32i)3-s + (−0.479 + 0.140i)4-s + (0.376 − 0.241i)5-s + (1.04 + 0.669i)6-s + (0.100 + 0.699i)7-s + (−0.146 − 0.321i)8-s + (−0.294 − 2.04i)9-s + (0.207 + 0.238i)10-s + (0.310 − 0.199i)11-s + (−0.363 + 0.796i)12-s + (0.0217 − 0.0475i)13-s + (−0.479 + 0.140i)14-s + (0.111 − 0.775i)15-s + (0.210 − 0.135i)16-s + (1.33 + 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.889 + 0.457i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.889 + 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30408 - 0.558143i\)
\(L(\frac12)\) \(\approx\) \(2.30408 - 0.558143i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 - 0.989i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (6.98 - 4.26i)T \)
good3 \( 1 + (-1.98 + 2.29i)T + (-0.426 - 2.96i)T^{2} \)
7 \( 1 + (-0.266 - 1.85i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-1.03 + 0.662i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-0.0783 + 0.171i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-5.49 - 1.61i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-0.662 + 4.60i)T + (-18.2 - 5.35i)T^{2} \)
23 \( 1 + (0.455 - 0.526i)T + (-3.27 - 22.7i)T^{2} \)
29 \( 1 + 5.36T + 29T^{2} \)
31 \( 1 + (-0.884 - 1.93i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + 3.17T + 37T^{2} \)
41 \( 1 + (6.34 + 1.86i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-1.78 - 0.525i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (-0.432 + 0.498i)T + (-6.68 - 46.5i)T^{2} \)
53 \( 1 + (4.08 - 1.20i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-3.01 - 6.59i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-8.79 - 5.65i)T + (25.3 + 55.4i)T^{2} \)
71 \( 1 + (9.08 - 2.66i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (6.00 + 3.86i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (3.72 - 8.16i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-0.779 + 0.501i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-6.72 - 7.76i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 - 7.10T + 97T^{2} \)
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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

−10.09936108081858590561573002534, −8.992425830508583328104588426016, −8.727656294457641868495826866504, −7.74901973103508917753640935300, −7.08909655673101663615306835805, −6.12809016339960088048718030758, −5.29854954432289406317884928521, −3.64700275956054716091622011070, −2.57012219550029679630323049570, −1.29503232476770142323656538724, 1.79776273945792078078920658721, 3.16432949070564482777565275462, 3.74687139286436199282467727477, 4.69778062851673152931961302324, 5.72870418330994723596799259757, 7.37978315932704732599807624896, 8.220763105661574901358094915884, 9.174270589545402758359176028541, 9.955867009043522667285941278678, 10.19783194717272326365712104083

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