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Properties

Label 2-850-17.13-c1-0-21
Degree $2$
Conductor $850$
Sign $0.986 - 0.161i$
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 + (0.750 + 0.750i)3-s − 4-s + (−0.750 + 0.750i)6-s + (1.70 − 1.70i)7-s − i·8-s − 1.87i·9-s + (3.64 − 3.64i)11-s + (−0.750 − 0.750i)12-s − 5.59·13-s + (1.70 + 1.70i)14-s + 16-s + (3.02 + 2.79i)17-s + 1.87·18-s − 2.18i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.433 + 0.433i)3-s − 0.5·4-s + (−0.306 + 0.306i)6-s + (0.644 − 0.644i)7-s − 0.353i·8-s − 0.624i·9-s + (1.09 − 1.09i)11-s + (−0.216 − 0.216i)12-s − 1.55·13-s + (0.455 + 0.455i)14-s + 0.250·16-s + (0.734 + 0.678i)17-s + 0.441·18-s − 0.501i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.986 - 0.161i$
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: \((2,\ 850,\ (\ :1/2),\ 0.986 - 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78696 + 0.145272i\)
\(L(\frac12)\) \(\approx\) \(1.78696 + 0.145272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.02 - 2.79i)T \)
good3 \( 1 + (-0.750 - 0.750i)T + 3iT^{2} \)
7 \( 1 + (-1.70 + 1.70i)T - 7iT^{2} \)
11 \( 1 + (-3.64 + 3.64i)T - 11iT^{2} \)
13 \( 1 + 5.59T + 13T^{2} \)
19 \( 1 + 2.18iT - 19T^{2} \)
23 \( 1 + (-5.72 + 5.72i)T - 23iT^{2} \)
29 \( 1 + (6.78 + 6.78i)T + 29iT^{2} \)
31 \( 1 + (0.519 + 0.519i)T + 31iT^{2} \)
37 \( 1 + (-5.87 - 5.87i)T + 37iT^{2} \)
41 \( 1 + (2.95 - 2.95i)T - 41iT^{2} \)
43 \( 1 - 5.00iT - 43T^{2} \)
47 \( 1 + 1.03T + 47T^{2} \)
53 \( 1 - 8.18iT - 53T^{2} \)
59 \( 1 - 10.1iT - 59T^{2} \)
61 \( 1 + (-8.09 + 8.09i)T - 61iT^{2} \)
67 \( 1 + 4.50T + 67T^{2} \)
71 \( 1 + (-0.422 - 0.422i)T + 71iT^{2} \)
73 \( 1 + (-0.455 - 0.455i)T + 73iT^{2} \)
79 \( 1 + (-7.07 + 7.07i)T - 79iT^{2} \)
83 \( 1 - 14.2iT - 83T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + (-2.96 - 2.96i)T + 97iT^{2} \)
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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

−9.896169237355290596198793977733, −9.321563334388642989111734752085, −8.478757730401313738912612709925, −7.71318452542963594389996576597, −6.78083598248381300003022279646, −5.95977338208688638574335075678, −4.72146136039074961612075057179, −4.04277660744287522286746260657, −2.93030604012142012976722533628, −0.920639487078119589409230953784, 1.60959054997475281694399250820, 2.30267464468221703743131636931, 3.54226060895057347644589189690, 4.91424560891072265985558395901, 5.34905403493225862614503910108, 7.23063229363628237794208309187, 7.42445222290533739190325268876, 8.659537014935337248408988409155, 9.403816780080366299369511311300, 10.00373776352826983440226980571

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