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Properties

Label 2-850-85.37-c1-0-18
Degree $2$
Conductor $850$
Sign $-0.536 + 0.844i$
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  + (−0.923 − 0.382i)2-s + (0.432 − 0.288i)3-s + (0.707 + 0.707i)4-s + (−0.509 + 0.101i)6-s + (−3.10 + 0.618i)7-s + (−0.382 − 0.923i)8-s + (−1.04 + 2.52i)9-s + (0.0548 + 0.275i)11-s + (0.509 + 0.101i)12-s + 1.43·13-s + (3.10 + 0.618i)14-s + i·16-s + (0.813 − 4.04i)17-s + (1.93 − 1.93i)18-s + (−3.00 − 7.24i)19-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (0.249 − 0.166i)3-s + (0.353 + 0.353i)4-s + (−0.208 + 0.0414i)6-s + (−1.17 + 0.233i)7-s + (−0.135 − 0.326i)8-s + (−0.348 + 0.840i)9-s + (0.0165 + 0.0832i)11-s + (0.147 + 0.0292i)12-s + 0.397·13-s + (0.830 + 0.165i)14-s + 0.250i·16-s + (0.197 − 0.980i)17-s + (0.454 − 0.454i)18-s + (−0.688 − 1.66i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.536 + 0.844i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ -0.536 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301203 - 0.548108i\)
\(L(\frac12)\) \(\approx\) \(0.301203 - 0.548108i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 \)
17 \( 1 + (-0.813 + 4.04i)T \)
good3 \( 1 + (-0.432 + 0.288i)T + (1.14 - 2.77i)T^{2} \)
7 \( 1 + (3.10 - 0.618i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (-0.0548 - 0.275i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 - 1.43T + 13T^{2} \)
19 \( 1 + (3.00 + 7.24i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.71 + 4.06i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (2.28 + 3.42i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (-1.68 + 8.47i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (4.87 + 7.29i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (4.80 - 7.18i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (-6.28 + 2.60i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 2.58iT - 47T^{2} \)
53 \( 1 + (1.55 - 3.76i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.245 - 0.101i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-7.53 - 5.03i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (3.18 - 3.18i)T - 67iT^{2} \)
71 \( 1 + (5.40 + 1.07i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-2.68 - 0.534i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (-10.5 + 2.09i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (10.1 + 4.21i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (5.32 + 5.32i)T + 89iT^{2} \)
97 \( 1 + (11.7 + 2.34i)T + (89.6 + 37.1i)T^{2} \)
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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

−9.732831473883202865712274676819, −9.111147597681357200364404484459, −8.403108934051908014536919923032, −7.35787348182291094571937556034, −6.68193170041952655855181708041, −5.62977822687963505859523997551, −4.36127589058268523945869563572, −2.95747687702187767683908893163, −2.34380631608740167345251010564, −0.36659646714734383378932536328, 1.45266006781132231466310943185, 3.22540535050396438603145193568, 3.78388146471437347921362974704, 5.50866642029057117234515693237, 6.34162360875848160432869712217, 6.92796206575837863246451005719, 8.186424303028602752135018483039, 8.754750348726052193030527260145, 9.635929805493544761021176035437, 10.23422273527539348078323186240

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