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Properties

Label 2-670-67.37-c1-0-20
Degree $2$
Conductor $670$
Sign $-0.311 - 0.950i$
Analytic cond. $5.34997$
Root an. cond. $2.31300$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s − 3·3-s + (−0.499 + 0.866i)4-s − 5-s + (1.5 + 2.59i)6-s + (1 − 1.73i)7-s + 0.999·8-s + 6·9-s + (0.5 + 0.866i)10-s + (1.49 − 2.59i)12-s + (−3 − 5.19i)13-s − 1.99·14-s + 3·15-s + (−0.5 − 0.866i)16-s + (−1.5 − 2.59i)17-s + (−3 − 5.19i)18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s − 1.73·3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (0.612 + 1.06i)6-s + (0.377 − 0.654i)7-s + 0.353·8-s + 2·9-s + (0.158 + 0.273i)10-s + (0.433 − 0.749i)12-s + (−0.832 − 1.44i)13-s − 0.534·14-s + 0.774·15-s + (−0.125 − 0.216i)16-s + (−0.363 − 0.630i)17-s + (−0.707 − 1.22i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.311 - 0.950i$
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: \(1\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ -0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (8 - 1.73i)T \)
good3 \( 1 + 3T + 3T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 9T + 59T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (7.5 + 12.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.31743656886623029454615366391, −9.397505387607453764337411622249, −7.928320430496155723447027889584, −7.34201805398307987865435794782, −6.30484030872002415915316300177, −5.04706423752142826048461437808, −4.61524556361712065430856335129, −3.13370833949037840390314518547, −1.17496949893074748858140177876, 0, 1.82823849821297624023396428960, 4.22532156000322274052720519635, 4.94830563886166977831356339122, 5.80439346955403942548675054102, 6.69467179379962270705342280345, 7.26656054753454172653009808036, 8.521110211718170907828007986778, 9.364626699557933190839776245270, 10.45203458830192855848177078668

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