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Properties

Label 2-670-67.29-c1-0-5
Degree $2$
Conductor $670$
Sign $-0.667 - 0.744i$
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 + 3-s + (−0.499 − 0.866i)4-s − 5-s + (−0.5 + 0.866i)6-s + (2.5 + 4.33i)7-s + 0.999·8-s − 2·9-s + (0.5 − 0.866i)10-s + (−0.499 − 0.866i)12-s + (−1 + 1.73i)13-s − 5·14-s − 15-s + (−0.5 + 0.866i)16-s + (−2 + 3.46i)17-s + (1 − 1.73i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + 0.577·3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (−0.204 + 0.353i)6-s + (0.944 + 1.63i)7-s + 0.353·8-s − 0.666·9-s + (0.158 − 0.273i)10-s + (−0.144 − 0.249i)12-s + (−0.277 + 0.480i)13-s − 1.33·14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.485 + 0.840i)17-s + (0.235 − 0.408i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.667 - 0.744i$
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: \((2,\ 670,\ (\ :1/2),\ -0.667 - 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479709 + 1.07380i\)
\(L(\frac12)\) \(\approx\) \(0.479709 + 1.07380i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.5 + 6.06i)T \)
good3 \( 1 - T + 3T^{2} \)
7 \( 1 + (-2.5 - 4.33i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.5 - 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.5 - 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.5 + 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.00240837778025657177719571051, −9.299710999864686287774418806028, −9.154067369767058313614863434336, −8.165507236354733575843488136240, −7.70024231793119233254013421234, −6.36986244034035048614481318142, −5.45555914718782109440634505008, −4.62085334030757497228015576089, −3.03708655440262211666087944815, −1.90854467999987550260178269311, 0.65829431911630559611574131877, 2.21786478917475793254967020430, 3.55689517804545548531309016809, 4.23421176613325180122973279167, 5.44760817411127275314158423486, 7.13440399420706268152387043263, 7.77654896079089893249099301211, 8.345033032951210071439454739946, 9.358184501502686784865929722887, 10.43744741422891660864341674634

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