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Properties

Label 2-670-67.25-c1-0-17
Degree $2$
Conductor $670$
Sign $-0.926 + 0.375i$
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  + (−0.142 − 0.989i)2-s + (1.91 − 2.20i)3-s + (−0.959 + 0.281i)4-s + (−0.841 + 0.540i)5-s + (−2.45 − 1.57i)6-s + (−0.139 − 0.968i)7-s + (0.415 + 0.909i)8-s + (−0.785 − 5.46i)9-s + (0.654 + 0.755i)10-s + (3.02 − 1.94i)11-s + (−1.21 + 2.65i)12-s + (1.24 − 2.72i)13-s + (−0.938 + 0.275i)14-s + (−0.415 + 2.88i)15-s + (0.841 − 0.540i)16-s + (−1.93 − 0.567i)17-s + ⋯
L(s)  = 1  + (−0.100 − 0.699i)2-s + (1.10 − 1.27i)3-s + (−0.479 + 0.140i)4-s + (−0.376 + 0.241i)5-s + (−1.00 − 0.644i)6-s + (−0.0526 − 0.366i)7-s + (0.146 + 0.321i)8-s + (−0.261 − 1.82i)9-s + (0.207 + 0.238i)10-s + (0.911 − 0.585i)11-s + (−0.349 + 0.766i)12-s + (0.345 − 0.755i)13-s + (−0.250 + 0.0736i)14-s + (−0.107 + 0.745i)15-s + (0.210 − 0.135i)16-s + (−0.468 − 0.137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.926 + 0.375i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ -0.926 + 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337057 - 1.72903i\)
\(L(\frac12)\) \(\approx\) \(0.337057 - 1.72903i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 + 0.989i)T \)
5 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (-1.32 + 8.07i)T \)
good3 \( 1 + (-1.91 + 2.20i)T + (-0.426 - 2.96i)T^{2} \)
7 \( 1 + (0.139 + 0.968i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-3.02 + 1.94i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-1.24 + 2.72i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (1.93 + 0.567i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-0.0814 + 0.566i)T + (-18.2 - 5.35i)T^{2} \)
23 \( 1 + (3.33 - 3.84i)T + (-3.27 - 22.7i)T^{2} \)
29 \( 1 + 4.86T + 29T^{2} \)
31 \( 1 + (-0.515 - 1.12i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 + (-8.27 - 2.42i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (0.131 + 0.0385i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (-0.316 + 0.364i)T + (-6.68 - 46.5i)T^{2} \)
53 \( 1 + (-11.0 + 3.24i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-4.09 - 8.97i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-7.58 - 4.87i)T + (25.3 + 55.4i)T^{2} \)
71 \( 1 + (-2.31 + 0.679i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (2.87 + 1.84i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (3.84 - 8.40i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-8.28 + 5.32i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (0.873 + 1.00i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 - 5.06T + 97T^{2} \)
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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

−10.11306026512592418634631694801, −8.997859297973876694075950044616, −8.511978442211711158018162508261, −7.57366389721690915412890486476, −6.93048986234989726528323468895, −5.76438689072387920568027199163, −3.92649312099488907083048502825, −3.29448538907821429586493900459, −2.12146198526090335093757506140, −0.903831253630198294034498736471, 2.17803213869173373573420337103, 3.83175382595959791096640709138, 4.17282271006247833748805408983, 5.27474979031401007747239219249, 6.53983042433735775548276899162, 7.57166528990158457145831941521, 8.675108170553987085358166076522, 8.919446802138832810729876633697, 9.713257985548038800165312484485, 10.54695648950605720476680622594

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