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Properties

Label 2-670-67.37-c1-0-7
Degree $2$
Conductor $670$
Sign $0.431 - 0.902i$
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 − 2.51·3-s + (−0.499 + 0.866i)4-s + 5-s + (−1.25 − 2.17i)6-s + (−0.0966 + 0.167i)7-s − 0.999·8-s + 3.32·9-s + (0.5 + 0.866i)10-s + (2.66 − 4.60i)11-s + (1.25 − 2.17i)12-s + (−0.0966 − 0.167i)13-s − 0.193·14-s − 2.51·15-s + (−0.5 − 0.866i)16-s + (2.91 + 5.05i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 1.45·3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (−0.513 − 0.888i)6-s + (−0.0365 + 0.0632i)7-s − 0.353·8-s + 1.10·9-s + (0.158 + 0.273i)10-s + (0.802 − 1.38i)11-s + (0.362 − 0.628i)12-s + (−0.0267 − 0.0464i)13-s − 0.0516·14-s − 0.649·15-s + (−0.125 − 0.216i)16-s + (0.707 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.431 - 0.902i$
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: \((2,\ 670,\ (\ :1/2),\ 0.431 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.999104 + 0.629862i\)
\(L(\frac12)\) \(\approx\) \(0.999104 + 0.629862i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.98 + 4.27i)T \)
good3 \( 1 + 2.51T + 3T^{2} \)
7 \( 1 + (0.0966 - 0.167i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.66 + 4.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0966 + 0.167i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.91 - 5.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.903 - 1.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.563 + 0.976i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.485 - 0.841i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.34 - 9.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.339 - 0.588i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 2.51T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.83T + 53T^{2} \)
59 \( 1 - 5.73T + 59T^{2} \)
61 \( 1 + (0.320 + 0.555i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-4.75 + 8.23i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.91 - 3.32i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.292 + 0.506i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.36 + 2.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.67T + 89T^{2} \)
97 \( 1 + (7.43 + 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.81808833643089592086566043022, −9.941971996518892747850357579184, −8.849318853727801245468446593256, −7.933267192688254635034077993756, −6.67622722856491977706585308044, −6.00506621903753596669238991270, −5.63126030656632741817659312445, −4.47481322137972208428962411716, −3.30173385619897281444680134255, −1.11153252346691064056518930275, 0.893467139065450522110191279784, 2.33726156535412897149834713278, 4.00775858484400262491542643551, 4.96358356936324905597102788360, 5.59802419615995192348886035554, 6.66276047275552511670861110896, 7.31490218406161673354292770100, 9.054050354269939785126508991771, 9.811799225200701102243095428983, 10.43657274872141644335256694054

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