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Properties

Label 2-670-67.37-c1-0-17
Degree $2$
Conductor $670$
Sign $0.797 - 0.603i$
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 + 2.85·3-s + (−0.499 + 0.866i)4-s − 5-s + (1.42 + 2.47i)6-s + (2.24 − 3.89i)7-s − 0.999·8-s + 5.15·9-s + (−0.5 − 0.866i)10-s + (−1.57 + 2.73i)11-s + (−1.42 + 2.47i)12-s + (0.503 + 0.872i)13-s + 4.49·14-s − 2.85·15-s + (−0.5 − 0.866i)16-s + (2.38 + 4.13i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + 1.64·3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (0.583 + 1.00i)6-s + (0.849 − 1.47i)7-s − 0.353·8-s + 1.71·9-s + (−0.158 − 0.273i)10-s + (−0.476 + 0.825i)11-s + (−0.412 + 0.714i)12-s + (0.139 + 0.241i)13-s + 1.20·14-s − 0.737·15-s + (−0.125 − 0.216i)16-s + (0.579 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.797 - 0.603i$
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: \((2,\ 670,\ (\ :1/2),\ 0.797 - 0.603i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.81806 + 0.945971i\)
\(L(\frac12)\) \(\approx\) \(2.81806 + 0.945971i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.23 - 7.00i)T \)
good3 \( 1 - 2.85T + 3T^{2} \)
7 \( 1 + (-2.24 + 3.89i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.57 - 2.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.503 - 0.872i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.38 - 4.13i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.11 - 1.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.98 + 3.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.41 + 4.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.72 + 6.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.280 - 0.485i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.45 - 4.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 9.06T + 43T^{2} \)
47 \( 1 + (6.34 - 10.9i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.55T + 53T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 + (6.68 + 11.5i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (1.95 - 3.37i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.32 + 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.08 - 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.82 + 11.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.52T + 89T^{2} \)
97 \( 1 + (-0.642 - 1.11i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.26178568228975012092108577466, −9.737112323058590139938461244471, −8.345151665327645720779015141579, −7.953344205734067786209411069402, −7.50269917247059898241241566586, −6.41472515301266769925868217813, −4.60957609352618527769393833792, −4.17789918273478288837616494338, −3.17181906190928789335550351826, −1.68514773957835942276696623959, 1.68339685757953089835133830933, 2.90057143995188526752239465169, 3.27375658601651613081191096883, 4.78679670688799121544320323391, 5.55399827647870294318172922246, 7.17316680234389998987675824011, 8.320898004969946455144241763807, 8.521843845007969873112338486312, 9.390380461901075718139219966776, 10.33901398655387574056769865058

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