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Properties

Label 2-850-17.13-c1-0-24
Degree $2$
Conductor $850$
Sign $-0.560 + 0.828i$
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  − i·2-s + (0.511 + 0.511i)3-s − 4-s + (0.511 − 0.511i)6-s + (2.15 − 2.15i)7-s + i·8-s − 2.47i·9-s + (0.0827 − 0.0827i)11-s + (−0.511 − 0.511i)12-s − 5.60·13-s + (−2.15 − 2.15i)14-s + 16-s + (−4.11 + 0.280i)17-s − 2.47·18-s − 3.74i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.295 + 0.295i)3-s − 0.5·4-s + (0.208 − 0.208i)6-s + (0.814 − 0.814i)7-s + 0.353i·8-s − 0.825i·9-s + (0.0249 − 0.0249i)11-s + (−0.147 − 0.147i)12-s − 1.55·13-s + (−0.576 − 0.576i)14-s + 0.250·16-s + (−0.997 + 0.0681i)17-s − 0.583·18-s − 0.860i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.560 + 0.828i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ -0.560 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.667714 - 1.25778i\)
\(L(\frac12)\) \(\approx\) \(0.667714 - 1.25778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
17 \( 1 + (4.11 - 0.280i)T \)
good3 \( 1 + (-0.511 - 0.511i)T + 3iT^{2} \)
7 \( 1 + (-2.15 + 2.15i)T - 7iT^{2} \)
11 \( 1 + (-0.0827 + 0.0827i)T - 11iT^{2} \)
13 \( 1 + 5.60T + 13T^{2} \)
19 \( 1 + 3.74iT - 19T^{2} \)
23 \( 1 + (-6.08 + 6.08i)T - 23iT^{2} \)
29 \( 1 + (-2.85 - 2.85i)T + 29iT^{2} \)
31 \( 1 + (-4.90 - 4.90i)T + 31iT^{2} \)
37 \( 1 + (6.47 + 6.47i)T + 37iT^{2} \)
41 \( 1 + (-4.68 + 4.68i)T - 41iT^{2} \)
43 \( 1 - 0.0451iT - 43T^{2} \)
47 \( 1 + 9.81T + 47T^{2} \)
53 \( 1 + 4.70iT - 53T^{2} \)
59 \( 1 + 7.16iT - 59T^{2} \)
61 \( 1 + (0.584 - 0.584i)T - 61iT^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 + (-3.67 - 3.67i)T + 71iT^{2} \)
73 \( 1 + (-4.66 - 4.66i)T + 73iT^{2} \)
79 \( 1 + (-1.29 + 1.29i)T - 79iT^{2} \)
83 \( 1 - 0.941iT - 83T^{2} \)
89 \( 1 - 5.94T + 89T^{2} \)
97 \( 1 + (-5.10 - 5.10i)T + 97iT^{2} \)
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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

−10.02868308850990638475257619371, −9.064145781799397826982346362687, −8.548098043747133624554580153791, −7.29714093317534881956921519160, −6.64124126665950556674820939234, −4.84012878820969243881558970545, −4.61506150442047538041464461444, −3.31523671422301676297034543917, −2.27898010850079588473909589455, −0.67111879960464566237480387320, 1.81179423521251828124452083235, 2.87188999130142981277460923446, 4.66142439146520248670385284092, 5.05564025549658532478332455868, 6.16425194079526517316869196517, 7.27863222730071863898631320435, 7.85394723600980023269643422940, 8.581343405740405005314907484433, 9.436269128116781154174986408615, 10.30724799019630208778939476836

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