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Properties

Label 2-670-335.104-c1-0-11
Degree $2$
Conductor $670$
Sign $0.426 - 0.904i$
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 + 2.35i·3-s + (0.499 − 0.866i)4-s + (−1.42 − 1.71i)5-s + (−1.17 − 2.04i)6-s + (3.13 + 1.81i)7-s + 0.999i·8-s − 2.55·9-s + (2.09 + 0.774i)10-s + (2.18 − 3.77i)11-s + (2.04 + 1.17i)12-s + (1.99 − 1.15i)13-s − 3.62·14-s + (4.05 − 3.36i)15-s + (−0.5 − 0.866i)16-s + (2.39 − 1.38i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + 1.36i·3-s + (0.249 − 0.433i)4-s + (−0.639 − 0.769i)5-s + (−0.481 − 0.833i)6-s + (1.18 + 0.684i)7-s + 0.353i·8-s − 0.851·9-s + (0.663 + 0.244i)10-s + (0.657 − 1.13i)11-s + (0.589 + 0.340i)12-s + (0.552 − 0.319i)13-s − 0.967·14-s + (1.04 − 0.869i)15-s + (−0.125 − 0.216i)16-s + (0.580 − 0.335i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.426 - 0.904i$
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: \((2,\ 670,\ (\ :1/2),\ 0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04385 + 0.662041i\)
\(L(\frac12)\) \(\approx\) \(1.04385 + 0.662041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 + 1.71i)T \)
67 \( 1 + (-2.67 - 7.73i)T \)
good3 \( 1 - 2.35iT - 3T^{2} \)
7 \( 1 + (-3.13 - 1.81i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.18 + 3.77i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.99 + 1.15i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.39 + 1.38i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.148 - 0.257i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.34 + 3.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.50 - 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.16 + 3.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.22 + 1.28i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.97 + 5.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 7.21iT - 43T^{2} \)
47 \( 1 + (10.1 + 5.88i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.4iT - 53T^{2} \)
59 \( 1 + 4.01T + 59T^{2} \)
61 \( 1 + (-3.45 - 5.98i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-2.20 + 3.82i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.2 - 5.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.66 + 6.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.51 - 2.02i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.67T + 89T^{2} \)
97 \( 1 + (-10.8 + 6.24i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.74193598563870046653496004587, −9.555308213709591596554189001352, −8.772057252212936452463278867224, −8.509109669938270150148350923749, −7.48216780049261134092175992879, −5.90293001574324978800379174288, −5.16813566571478831936948875636, −4.35612152194508573931780018534, −3.21098828132367313849522663675, −1.14237046846592241619810391382, 1.16336746903362642282217574858, 2.00966440577257953579378944712, 3.52193041593918337204837999277, 4.66138533042670695484772695190, 6.43745732507722026579925585734, 7.05995089304755757916661773159, 7.74955476500374931869999643073, 8.217804195605817429191792519931, 9.518294252306365871626836890827, 10.52859672190529839803910305618

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