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Properties

Label 2-670-67.29-c1-0-7
Degree $2$
Conductor $670$
Sign $0.569 - 0.822i$
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 − 1.44·3-s + (−0.499 − 0.866i)4-s − 5-s + (0.724 − 1.25i)6-s + (0.224 + 0.389i)7-s + 0.999·8-s − 0.898·9-s + (0.5 − 0.866i)10-s + (0.724 + 1.25i)12-s + (2 − 3.46i)13-s − 0.449·14-s + 1.44·15-s + (−0.5 + 0.866i)16-s + (0.224 − 0.389i)17-s + (0.449 − 0.778i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s − 0.836·3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (0.295 − 0.512i)6-s + (0.0849 + 0.147i)7-s + 0.353·8-s − 0.299·9-s + (0.158 − 0.273i)10-s + (0.209 + 0.362i)12-s + (0.554 − 0.960i)13-s − 0.120·14-s + 0.374·15-s + (−0.125 + 0.216i)16-s + (0.0545 − 0.0944i)17-s + (0.105 − 0.183i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.569 - 0.822i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.569 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.654983 + 0.343085i\)
\(L(\frac12)\) \(\approx\) \(0.654983 + 0.343085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.17 - 3.94i)T \)
good3 \( 1 + 1.44T + 3T^{2} \)
7 \( 1 + (-0.224 - 0.389i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.224 + 0.389i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.44 - 2.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.72 - 4.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.94 + 3.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.94 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-3.67 - 6.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7T + 53T^{2} \)
59 \( 1 - 2.89T + 59T^{2} \)
61 \( 1 + (-0.224 + 0.389i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-7.17 - 12.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.77 + 3.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.17 + 2.03i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (-1.89 + 3.28i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.78348088403573331495899536883, −9.830680362541901843768566436519, −8.619568860285913425045412857410, −8.149575083931492048456559567054, −7.03438739450609445139699751051, −6.13405628585664837971910124744, −5.44066764208940040389596330065, −4.45243787328466640076244354403, −2.99955268417083510243135755704, −0.905324468767731353541492615310, 0.72511342655021426940638663774, 2.41146332732967891324347949785, 3.83243658174993329628210239349, 4.70146158013171872722216735303, 5.90829046579464146091190292766, 6.80636625732174085987711525350, 7.87954906209594308692364039718, 8.769170432997879302155205538055, 9.568974455762179851291038682820, 10.70940443588489164951004781800

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