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Properties

Label 2-850-85.12-c1-0-13
Degree $2$
Conductor $850$
Sign $0.994 + 0.105i$
Analytic cond. $6.78728$
Root an. cond. $2.60524$
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  + (−0.382 + 0.923i)2-s + (−0.567 + 2.85i)3-s + (−0.707 − 0.707i)4-s + (−2.41 − 1.61i)6-s + (−3.60 − 2.40i)7-s + (0.923 − 0.382i)8-s + (−5.03 − 2.08i)9-s + (−0.113 + 0.170i)11-s + (2.41 − 1.61i)12-s − 3.32·13-s + (3.60 − 2.40i)14-s + i·16-s + (4.05 + 0.761i)17-s + (3.85 − 3.85i)18-s + (1.76 − 0.729i)19-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (−0.327 + 1.64i)3-s + (−0.353 − 0.353i)4-s + (−0.986 − 0.659i)6-s + (−1.36 − 0.910i)7-s + (0.326 − 0.135i)8-s + (−1.67 − 0.695i)9-s + (−0.0343 + 0.0513i)11-s + (0.697 − 0.466i)12-s − 0.923·13-s + (0.963 − 0.643i)14-s + 0.250i·16-s + (0.982 + 0.184i)17-s + (0.908 − 0.908i)18-s + (0.403 − 0.167i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.994 + 0.105i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ 0.994 + 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.451150 - 0.0239362i\)
\(L(\frac12)\) \(\approx\) \(0.451150 - 0.0239362i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 \)
17 \( 1 + (-4.05 - 0.761i)T \)
good3 \( 1 + (0.567 - 2.85i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (3.60 + 2.40i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (0.113 - 0.170i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + 3.32T + 13T^{2} \)
19 \( 1 + (-1.76 + 0.729i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.769 + 0.153i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-8.95 - 1.78i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (4.85 + 7.26i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (-1.88 - 0.374i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (3.42 - 0.682i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (2.60 + 6.29i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 11.8iT - 47T^{2} \)
53 \( 1 + (4.78 + 1.97i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.58 - 3.81i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.209 - 1.05i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-5.49 + 5.49i)T - 67iT^{2} \)
71 \( 1 + (1.33 - 0.892i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-0.677 + 0.452i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (10.3 + 6.88i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-0.269 + 0.650i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (3.70 + 3.70i)T + 89iT^{2} \)
97 \( 1 + (0.124 - 0.0834i)T + (37.1 - 89.6i)T^{2} \)
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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

−10.04840074768538533562769633934, −9.625057851939711604608791267963, −8.732195016204312741206035036603, −7.52247157310671172684127863944, −6.68255214991630524970430679637, −5.69022163170318472863001571510, −4.86932847019500070346631145408, −3.92443068860585512214208154773, −3.11818862907535521845264012272, −0.27974926916691776952022264735, 1.19583869276393190277515027660, 2.56031316805578467828222110629, 3.16057735503618230763323069177, 5.08535889889313820511632622543, 6.04363807512117715649180958557, 6.79022048630667136972888252063, 7.62452829100637807521515551650, 8.460236291608044343246667899756, 9.451410513292605134779543800520, 10.09179652712484264230762781049

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