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Properties

Label 2-670-67.37-c1-0-19
Degree $2$
Conductor $670$
Sign $0.404 + 0.914i$
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 + 2.41·3-s + (−0.499 + 0.866i)4-s + 5-s + (−1.20 − 2.09i)6-s + (0.707 − 1.22i)7-s + 0.999·8-s + 2.82·9-s + (−0.5 − 0.866i)10-s + (1 − 1.73i)11-s + (−1.20 + 2.09i)12-s + (−2.56 − 4.44i)13-s − 1.41·14-s + 2.41·15-s + (−0.5 − 0.866i)16-s + (−2.52 − 4.37i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + 1.39·3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (−0.492 − 0.853i)6-s + (0.267 − 0.462i)7-s + 0.353·8-s + 0.942·9-s + (−0.158 − 0.273i)10-s + (0.301 − 0.522i)11-s + (−0.348 + 0.603i)12-s + (−0.712 − 1.23i)13-s − 0.377·14-s + 0.623·15-s + (−0.125 − 0.216i)16-s + (−0.612 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.404 + 0.914i$
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: \((2,\ 670,\ (\ :1/2),\ 0.404 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76240 - 1.14765i\)
\(L(\frac12)\) \(\approx\) \(1.76240 - 1.14765i\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.78 - 2.51i)T \)
good3 \( 1 - 2.41T + 3T^{2} \)
7 \( 1 + (-0.707 + 1.22i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.56 + 4.44i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.52 + 4.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.81 - 4.87i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.33 - 7.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.15 + 3.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.37 + 5.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.73 - 6.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.487 - 0.845i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.33T + 43T^{2} \)
47 \( 1 + (4.68 - 8.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.50T + 53T^{2} \)
59 \( 1 - 3.63T + 59T^{2} \)
61 \( 1 + (-6.10 - 10.5i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-3.02 + 5.23i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.35 + 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.46 - 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.78 + 10.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + (-1.58 - 2.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.848651907347394229768945461054, −9.746902129516436019027587790483, −8.693623197935378044587509172780, −7.86764496768096153325913409741, −7.35476945547095570994031629707, −5.76927187038014214609291468805, −4.51047944065775614055367791683, −3.25463462322324733161224498306, −2.68468143466536333326136244429, −1.24321718477069249939475477949, 1.81927663956023760616326068978, 2.71622001213851639503628025041, 4.25840440961854156230377339547, 5.12942819243375682552549200913, 6.67725494370970020589645143576, 7.06287318231243341931028073585, 8.404792748899954163507639508621, 8.805728252456782160889241980642, 9.427903083720468421331110477667, 10.28683882366519855126764694786

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