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Properties

Label 2-670-67.29-c1-0-18
Degree $2$
Conductor $670$
Sign $-0.396 + 0.917i$
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 − 0.146·3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.0731 + 0.126i)6-s + (0.573 + 0.992i)7-s − 0.999·8-s − 2.97·9-s + (0.5 − 0.866i)10-s + (−2 − 3.46i)11-s + (0.0731 + 0.126i)12-s + (2.34 − 4.05i)13-s + 1.14·14-s − 0.146·15-s + (−0.5 + 0.866i)16-s + (0.916 − 1.58i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s − 0.0845·3-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (−0.0298 + 0.0517i)6-s + (0.216 + 0.375i)7-s − 0.353·8-s − 0.992·9-s + (0.158 − 0.273i)10-s + (−0.603 − 1.04i)11-s + (0.0211 + 0.0365i)12-s + (0.649 − 1.12i)13-s + 0.306·14-s − 0.0377·15-s + (−0.125 + 0.216i)16-s + (0.222 − 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.396 + 0.917i$
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: \((2,\ 670,\ (\ :1/2),\ -0.396 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.867035 - 1.31904i\)
\(L(\frac12)\) \(\approx\) \(0.867035 - 1.31904i\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.80 + 2.45i)T \)
good3 \( 1 + 0.146T + 3T^{2} \)
7 \( 1 + (-0.573 - 0.992i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.34 + 4.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.916 + 1.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.83 + 4.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.57 + 2.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.68 + 6.38i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.26 - 7.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.69 + 2.93i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.98 - 5.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-4.42 - 7.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 + 0.292T + 59T^{2} \)
61 \( 1 + (5.20 - 9.02i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (0.876 + 1.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.06 + 7.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.02 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.36 - 5.82i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.664T + 89T^{2} \)
97 \( 1 + (-3.97 + 6.89i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.54518095881598336952128796245, −9.391332899304660522883126610929, −8.635560659685984249605895501299, −7.83709269093935278790708080566, −6.25894155301614015386842033840, −5.61385733124902559763466653692, −4.87350520211993468108619792551, −3.17497422151215613540859571721, −2.67697170400291633178365338819, −0.77562865053076373122657692367, 1.82406914291028556060649460825, 3.34338176373669104276474895281, 4.47468948498605609140933374904, 5.47721960319881564522440034835, 6.20103557905433481239323330583, 7.25652166302252186620840958623, 8.007192312961461399405936348826, 9.005895980873595216834589029103, 9.800586959926668012601596272744, 10.81347159284979642821761687512

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