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Properties

Label 2-670-67.29-c1-0-4
Degree $2$
Conductor $670$
Sign $0.466 - 0.884i$
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 − 2.10·3-s + (−0.499 − 0.866i)4-s + 5-s + (−1.05 + 1.82i)6-s + (1.55 + 2.68i)7-s − 0.999·8-s + 1.42·9-s + (0.5 − 0.866i)10-s + (−2 − 3.46i)11-s + (1.05 + 1.82i)12-s + (−1.81 + 3.14i)13-s + 3.10·14-s − 2.10·15-s + (−0.5 + 0.866i)16-s + (−2.26 + 3.91i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s − 1.21·3-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (−0.429 + 0.743i)6-s + (0.586 + 1.01i)7-s − 0.353·8-s + 0.473·9-s + (0.158 − 0.273i)10-s + (−0.603 − 1.04i)11-s + (0.303 + 0.525i)12-s + (−0.503 + 0.871i)13-s + 0.829·14-s − 0.542·15-s + (−0.125 + 0.216i)16-s + (−0.548 + 0.950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.466 - 0.884i$
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: \((2,\ 670,\ (\ :1/2),\ 0.466 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692797 + 0.417741i\)
\(L(\frac12)\) \(\approx\) \(0.692797 + 0.417741i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.59 - 3.05i)T \)
good3 \( 1 + 2.10T + 3T^{2} \)
7 \( 1 + (-1.55 - 2.68i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.81 - 3.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.26 - 3.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.52 - 6.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.55 + 4.41i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.62 - 8.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.864 + 1.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.41 - 7.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.789 - 1.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-3.44 - 5.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + 4.20T + 59T^{2} \)
61 \( 1 + (5.94 - 10.2i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (7.96 + 13.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.84 + 4.91i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.44 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.25 - 14.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (0.421 - 0.730i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.65198399569055110272948700686, −10.36102691915819532363212744590, −8.848899843594143741406222487415, −8.419081247235697010024043431200, −6.63992963691494287918235008700, −5.94387245563187383428294648143, −5.27851675437723944315602039570, −4.41445351652433965560718690218, −2.80504400255658514049715686250, −1.59216395561569687032276779444, 0.44456184679367295362881142514, 2.52101476111860323610630110533, 4.36679221018499154479145937131, 4.99723605396183947142471410691, 5.66289985292517535894719000144, 7.07213453897463531103423809906, 7.12880981799745885206225888284, 8.454187215042786822485914301793, 9.684861964826398652846983816266, 10.50490872609738889677862068267

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