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Properties

Label 2-670-67.37-c1-0-4
Degree $2$
Conductor $670$
Sign $0.854 - 0.519i$
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 − 0.414·3-s + (−0.499 + 0.866i)4-s + 5-s + (0.207 + 0.358i)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 + (0.207 − 0.358i)12-s + (2.03 + 3.52i)13-s + 1.41·14-s − 0.414·15-s + (−0.5 − 0.866i)16-s + (−0.732 − 1.26i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s − 0.239·3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (0.0845 + 0.146i)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 + (0.0597 − 0.103i)12-s + (0.564 + 0.978i)13-s + 0.377·14-s − 0.106·15-s + (−0.125 − 0.216i)16-s + (−0.177 − 0.307i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.854 - 0.519i$
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: \((2,\ 670,\ (\ :1/2),\ 0.854 - 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.965382 + 0.270737i\)
\(L(\frac12)\) \(\approx\) \(0.965382 + 0.270737i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.51 - 7.39i)T \)
good3 \( 1 + 0.414T + 3T^{2} \)
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 \( 1 + (-2.03 - 3.52i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.732 + 1.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.43 - 4.22i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.17 - 3.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.378 + 0.654i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.09 - 8.82i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.525 - 0.910i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.76 - 9.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + (-4.15 + 7.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.95T + 53T^{2} \)
59 \( 1 - 2.87T + 59T^{2} \)
61 \( 1 + (5.57 + 9.65i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-1.23 + 2.13i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.09 - 1.89i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.05 - 1.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.51 - 9.54i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + (8.30 + 14.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.71521263863961635567741709480, −9.613368992056199536870365821530, −9.001582096192772056034363587565, −8.334969627777817084342987198615, −7.01805993236272733156795159459, −6.01740107859908892978616249160, −5.24294079561468439579895368694, −3.76058838469394131436186771350, −2.78827808112207500303926822967, −1.40380806274590722764616378977, 0.67487024917539968447610818728, 2.54954252679572968705456585233, 3.99004931231215351136592614921, 5.29459849062977323504589582160, 5.95273703975755089083957447807, 6.89634138418898217746813850162, 7.73324086773697034556503365495, 8.819684888748683753827044502566, 9.356665656141292774367126910822, 10.52370698953993281907461806898

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