↑

Properties

Label 2-859-859.100-c1-0-28
Degree $2$
Conductor $859$
Sign $0.311 - 0.950i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 + 0.719i)2-s + (−0.554 + 0.803i)3-s + (0.223 − 0.324i)4-s + (0.478 + 0.692i)5-s + (0.181 − 1.49i)6-s + (0.117 + 0.969i)7-s + (0.299 − 2.46i)8-s + (0.726 + 1.91i)9-s + (−1.15 − 0.605i)10-s + (−0.555 − 4.57i)11-s + (0.136 + 0.359i)12-s + 2.58·13-s + (−0.858 − 1.24i)14-s − 0.821·15-s + (1.64 + 4.33i)16-s + (7.12 − 1.75i)17-s + ⋯
L(s)  = 1  + (−0.968 + 0.508i)2-s + (−0.320 + 0.463i)3-s + (0.111 − 0.162i)4-s + (0.213 + 0.309i)5-s + (0.0742 − 0.611i)6-s + (0.0444 + 0.366i)7-s + (0.105 − 0.872i)8-s + (0.242 + 0.638i)9-s + (−0.364 − 0.191i)10-s + (−0.167 − 1.37i)11-s + (0.0393 + 0.103i)12-s + 0.717·13-s + (−0.229 − 0.332i)14-s − 0.212·15-s + (0.410 + 1.08i)16-s + (1.72 − 0.425i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(859\)
Sign: $0.311 - 0.950i$
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: \((2,\ 859,\ (\ :1/2),\ 0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.732048 + 0.530512i\)
\(L(\frac12)\) \(\approx\) \(0.732048 + 0.530512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad859 \( 1 + (-22.1 - 19.2i)T \)
good2 \( 1 + (1.37 - 0.719i)T + (1.13 - 1.64i)T^{2} \)
3 \( 1 + (0.554 - 0.803i)T + (-1.06 - 2.80i)T^{2} \)
5 \( 1 + (-0.478 - 0.692i)T + (-1.77 + 4.67i)T^{2} \)
7 \( 1 + (-0.117 - 0.969i)T + (-6.79 + 1.67i)T^{2} \)
11 \( 1 + (0.555 + 4.57i)T + (-10.6 + 2.63i)T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
17 \( 1 + (-7.12 + 1.75i)T + (15.0 - 7.90i)T^{2} \)
19 \( 1 + 0.113T + 19T^{2} \)
23 \( 1 + (-1.35 - 0.709i)T + (13.0 + 18.9i)T^{2} \)
29 \( 1 + (-2.43 + 6.41i)T + (-21.7 - 19.2i)T^{2} \)
31 \( 1 + (-6.69 - 5.93i)T + (3.73 + 30.7i)T^{2} \)
37 \( 1 + (2.90 + 7.64i)T + (-27.6 + 24.5i)T^{2} \)
41 \( 1 + (0.748 - 1.08i)T + (-14.5 - 38.3i)T^{2} \)
43 \( 1 + 3.79T + 43T^{2} \)
47 \( 1 + (5.64 - 5.00i)T + (5.66 - 46.6i)T^{2} \)
53 \( 1 + (-8.70 - 2.14i)T + (46.9 + 24.6i)T^{2} \)
59 \( 1 + (-3.67 + 1.92i)T + (33.5 - 48.5i)T^{2} \)
61 \( 1 + 7.94T + 61T^{2} \)
67 \( 1 + (-0.121 - 0.320i)T + (-50.1 + 44.4i)T^{2} \)
71 \( 1 + (-12.1 - 2.98i)T + (62.8 + 32.9i)T^{2} \)
73 \( 1 + (1.35 - 11.1i)T + (-70.8 - 17.4i)T^{2} \)
79 \( 1 + (3.89 - 5.63i)T + (-28.0 - 73.8i)T^{2} \)
83 \( 1 + (-1.59 - 13.1i)T + (-80.5 + 19.8i)T^{2} \)
89 \( 1 + (5.66 + 1.39i)T + (78.8 + 41.3i)T^{2} \)
97 \( 1 + (9.81 - 2.41i)T + (85.8 - 45.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.20648082030122277013789406726, −9.547617975665443666959188560175, −8.390726567647743494759294978127, −8.165626726007702628712741199667, −7.03323395550288789295768303912, −6.04795639494841174694397677056, −5.31871583750931287454140306598, −3.97086455495975565485561509869, −2.89206635426169399097394116996, −0.950772013123686558563321334237, 0.988307189029650924524770206425, 1.70638433940474295705976126319, 3.33348652626468980786280989800, 4.72144524461135206433666750470, 5.64150002746572169873123010370, 6.75192261689519265804101820039, 7.59096865061430030013219150246, 8.437038786902136723517902288006, 9.338533644357968036090160856466, 10.06001126444314423468106498507

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