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Properties

Label 2-859-859.100-c1-0-20
Degree $2$
Conductor $859$
Sign $-0.535 - 0.844i$
Analytic cond. $6.85914$
Root an. cond. $2.61899$
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  + (−1.74 + 0.913i)2-s + (0.895 − 1.29i)3-s + (1.06 − 1.53i)4-s + (1.65 + 2.39i)5-s + (−0.373 + 3.07i)6-s + (0.380 + 3.13i)7-s + (0.0317 − 0.261i)8-s + (0.182 + 0.481i)9-s + (−5.07 − 2.66i)10-s + (0.704 + 5.80i)11-s + (−1.04 − 2.75i)12-s + 4.81·13-s + (−3.52 − 5.10i)14-s + 4.59·15-s + (1.50 + 3.97i)16-s + (2.31 − 0.569i)17-s + ⋯
L(s)  = 1  + (−1.23 + 0.646i)2-s + (0.517 − 0.749i)3-s + (0.530 − 0.767i)4-s + (0.740 + 1.07i)5-s + (−0.152 + 1.25i)6-s + (0.143 + 1.18i)7-s + (0.0112 − 0.0923i)8-s + (0.0608 + 0.160i)9-s + (−1.60 − 0.842i)10-s + (0.212 + 1.74i)11-s + (−0.301 − 0.794i)12-s + 1.33·13-s + (−0.941 − 1.36i)14-s + 1.18·15-s + (0.376 + 0.993i)16-s + (0.560 − 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(859\)
Sign: $-0.535 - 0.844i$
Analytic conductor: \(6.85914\)
Root analytic conductor: \(2.61899\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{859} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 859,\ (\ :1/2),\ -0.535 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526044 + 0.956253i\)
\(L(\frac12)\) \(\approx\) \(0.526044 + 0.956253i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad859 \( 1 + (-28.9 + 4.81i)T \)
good2 \( 1 + (1.74 - 0.913i)T + (1.13 - 1.64i)T^{2} \)
3 \( 1 + (-0.895 + 1.29i)T + (-1.06 - 2.80i)T^{2} \)
5 \( 1 + (-1.65 - 2.39i)T + (-1.77 + 4.67i)T^{2} \)
7 \( 1 + (-0.380 - 3.13i)T + (-6.79 + 1.67i)T^{2} \)
11 \( 1 + (-0.704 - 5.80i)T + (-10.6 + 2.63i)T^{2} \)
13 \( 1 - 4.81T + 13T^{2} \)
17 \( 1 + (-2.31 + 0.569i)T + (15.0 - 7.90i)T^{2} \)
19 \( 1 + 7.47T + 19T^{2} \)
23 \( 1 + (7.71 + 4.04i)T + (13.0 + 18.9i)T^{2} \)
29 \( 1 + (0.868 - 2.29i)T + (-21.7 - 19.2i)T^{2} \)
31 \( 1 + (4.83 + 4.28i)T + (3.73 + 30.7i)T^{2} \)
37 \( 1 + (3.69 + 9.73i)T + (-27.6 + 24.5i)T^{2} \)
41 \( 1 + (-0.0166 + 0.0241i)T + (-14.5 - 38.3i)T^{2} \)
43 \( 1 - 4.44T + 43T^{2} \)
47 \( 1 + (2.05 - 1.82i)T + (5.66 - 46.6i)T^{2} \)
53 \( 1 + (-12.8 - 3.15i)T + (46.9 + 24.6i)T^{2} \)
59 \( 1 + (1.97 - 1.03i)T + (33.5 - 48.5i)T^{2} \)
61 \( 1 + 4.58T + 61T^{2} \)
67 \( 1 + (-3.53 - 9.31i)T + (-50.1 + 44.4i)T^{2} \)
71 \( 1 + (1.06 + 0.262i)T + (62.8 + 32.9i)T^{2} \)
73 \( 1 + (-1.53 + 12.6i)T + (-70.8 - 17.4i)T^{2} \)
79 \( 1 + (-8.29 + 12.0i)T + (-28.0 - 73.8i)T^{2} \)
83 \( 1 + (-1.51 - 12.4i)T + (-80.5 + 19.8i)T^{2} \)
89 \( 1 + (-9.94 - 2.45i)T + (78.8 + 41.3i)T^{2} \)
97 \( 1 + (10.4 - 2.57i)T + (85.8 - 45.0i)T^{2} \)
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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

−10.36756912935808806691160557428, −9.337117417196622716537730836775, −8.731547676507771019855118929806, −7.925930982658214257066497879236, −7.18479173685287773956329470396, −6.43831822333205232055274244273, −5.82016345055178986116973267798, −4.04292617856214109910185525205, −2.21087807931073717447009988707, −1.95197154377199840093228171563, 0.77901535829072968551471643878, 1.66209432292130709683246955393, 3.43207331165476365403195112510, 4.08363007981693534921667201985, 5.48339554954764853165814889358, 6.40420844180803658389728537035, 8.035244951440745612243574875750, 8.560785439270702158241353107928, 8.997164265748004681053235121195, 9.894516776590575385987229550680

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