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Properties

Label 2-670-335.104-c1-0-23
Degree $2$
Conductor $670$
Sign $0.875 + 0.482i$
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  + (−0.866 + 0.5i)2-s + 3.40i·3-s + (0.499 − 0.866i)4-s + (1.60 − 1.56i)5-s + (−1.70 − 2.94i)6-s + (−3.81 − 2.20i)7-s + 0.999i·8-s − 8.57·9-s + (−0.604 + 2.15i)10-s + (2.26 − 3.92i)11-s + (2.94 + 1.70i)12-s + (0.389 − 0.225i)13-s + 4.40·14-s + (5.31 + 5.44i)15-s + (−0.5 − 0.866i)16-s + (2.11 − 1.21i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + 1.96i·3-s + (0.249 − 0.433i)4-s + (0.715 − 0.698i)5-s + (−0.694 − 1.20i)6-s + (−1.44 − 0.832i)7-s + 0.353i·8-s − 2.85·9-s + (−0.191 + 0.680i)10-s + (0.683 − 1.18i)11-s + (0.850 + 0.491i)12-s + (0.108 − 0.0624i)13-s + 1.17·14-s + (1.37 + 1.40i)15-s + (−0.125 − 0.216i)16-s + (0.512 − 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.875 + 0.482i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.875 + 0.482i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640574 - 0.164893i\)
\(L(\frac12)\) \(\approx\) \(0.640574 - 0.164893i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-1.60 + 1.56i)T \)
67 \( 1 + (-1.54 - 8.03i)T \)
good3 \( 1 - 3.40iT - 3T^{2} \)
7 \( 1 + (3.81 + 2.20i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.26 + 3.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.389 + 0.225i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.11 + 1.21i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.49 + 4.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.81 - 1.04i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.33 + 4.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.84 - 6.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.64 + 1.52i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.68 - 4.64i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 1.96iT - 43T^{2} \)
47 \( 1 + (7.39 + 4.27i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.26iT - 53T^{2} \)
59 \( 1 - 4.27T + 59T^{2} \)
61 \( 1 + (6.76 + 11.7i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (6.40 - 11.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.02 + 1.16i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.65 + 2.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.769 - 0.444i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.369T + 89T^{2} \)
97 \( 1 + (-4.39 + 2.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.06834023441707717207537240042, −9.675716327692288174034279269359, −8.968617031224067484752357467774, −8.368270557751042162532354700158, −6.60710417731271139200075223006, −5.92549122942827384277106801859, −4.98684704936686763666074773894, −3.87956430820052113818610262925, −3.01892434146549784741368509166, −0.41659080225246447550012703158, 1.61887165502486377542771403174, 2.37926223736399166133724428096, 3.34745153945252317379188269372, 5.88040859775422753841303349298, 6.30883198950232192987787803633, 7.00691989656725996445977554662, 7.81647720782630601521117408717, 8.918635916161256656130085685112, 9.586547001739791606223794764704, 10.50644012619355853988027548660

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