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Properties

Label 2-670-67.14-c1-0-11
Degree $2$
Conductor $670$
Sign $-0.510 - 0.860i$
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  + (0.415 + 0.909i)2-s + (2.41 + 1.55i)3-s + (−0.654 + 0.755i)4-s + (0.142 + 0.989i)5-s + (−0.408 + 2.83i)6-s + (1.65 + 3.61i)7-s + (−0.959 − 0.281i)8-s + (2.17 + 4.75i)9-s + (−0.841 + 0.540i)10-s + (−0.806 − 5.60i)11-s + (−2.75 + 0.808i)12-s + (3.63 − 1.06i)13-s + (−2.60 + 3.00i)14-s + (−1.19 + 2.60i)15-s + (−0.142 − 0.989i)16-s + (0.0234 + 0.0270i)17-s + ⋯
L(s)  = 1  + (0.293 + 0.643i)2-s + (1.39 + 0.895i)3-s + (−0.327 + 0.377i)4-s + (0.0636 + 0.442i)5-s + (−0.166 + 1.15i)6-s + (0.624 + 1.36i)7-s + (−0.339 − 0.0996i)8-s + (0.723 + 1.58i)9-s + (−0.266 + 0.170i)10-s + (−0.243 − 1.69i)11-s + (−0.794 + 0.233i)12-s + (1.00 − 0.295i)13-s + (−0.696 + 0.803i)14-s + (−0.307 + 0.673i)15-s + (−0.0355 − 0.247i)16-s + (0.00567 + 0.00655i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.510 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.510 - 0.860i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ -0.510 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39201 + 2.44397i\)
\(L(\frac12)\) \(\approx\) \(1.39201 + 2.44397i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 - 0.909i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (-7.96 - 1.88i)T \)
good3 \( 1 + (-2.41 - 1.55i)T + (1.24 + 2.72i)T^{2} \)
7 \( 1 + (-1.65 - 3.61i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (0.806 + 5.60i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (-3.63 + 1.06i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.0234 - 0.0270i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-2.78 + 6.09i)T + (-12.4 - 14.3i)T^{2} \)
23 \( 1 + (6.27 + 4.03i)T + (9.55 + 20.9i)T^{2} \)
29 \( 1 + 7.29T + 29T^{2} \)
31 \( 1 + (7.85 + 2.30i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 - 5.46T + 37T^{2} \)
41 \( 1 + (1.23 + 1.42i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-4.39 - 5.06i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (-7.57 - 4.86i)T + (19.5 + 42.7i)T^{2} \)
53 \( 1 + (8.37 - 9.66i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-2.51 - 0.738i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-0.821 + 5.71i)T + (-58.5 - 17.1i)T^{2} \)
71 \( 1 + (-1.21 + 1.39i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-0.773 + 5.38i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (11.4 - 3.37i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (0.298 + 2.07i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (8.42 - 5.41i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−10.92766566960969119173944264853, −9.455207329126805994062987294007, −8.952496204858625143596772989842, −8.301657766723978411330659066937, −7.67949720580642791119827629873, −6.05449117876430822315563700616, −5.48310906810948889014092176063, −4.17912717143832837832979070706, −3.21498893197599031976188529275, −2.44858751899682180489189083750, 1.49675150630414822381365856384, 1.91836956248161197509949023884, 3.71085552640175141428447058791, 4.08518580100358000330188946436, 5.57044047578740817278946876496, 7.12464627927735591713533338752, 7.61675828730118054866612386142, 8.371398119070593478367904538966, 9.492933317775322853845142128009, 10.01634989535089695225384068817

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