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Properties

Label 2-850-17.13-c1-0-19
Degree $2$
Conductor $850$
Sign $0.788 - 0.615i$
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 + (1 + i)3-s − 4-s + (−1 + i)6-s + (2 − 2i)7-s − i·8-s − i·9-s + (1 − i)11-s + (−1 − i)12-s + 6·13-s + (2 + 2i)14-s + 16-s + (−4 − i)17-s + 18-s − 4i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.577 + 0.577i)3-s − 0.5·4-s + (−0.408 + 0.408i)6-s + (0.755 − 0.755i)7-s − 0.353i·8-s − 0.333i·9-s + (0.301 − 0.301i)11-s + (−0.288 − 0.288i)12-s + 1.66·13-s + (0.534 + 0.534i)14-s + 0.250·16-s + (−0.970 − 0.242i)17-s + 0.235·18-s − 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.788 - 0.615i$
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: \((2,\ 850,\ (\ :1/2),\ 0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92913 + 0.663912i\)
\(L(\frac12)\) \(\approx\) \(1.92913 + 0.663912i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 + i)T \)
good3 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (-2 + 2i)T - 7iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 - 6T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + (2 + 2i)T + 29iT^{2} \)
31 \( 1 + (-6 - 6i)T + 31iT^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + (-1 + i)T - 41iT^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (4 - 4i)T - 61iT^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (4 + 4i)T + 71iT^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 + (8 - 8i)T - 79iT^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-5 - 5i)T + 97iT^{2} \)
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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

−10.18333568577029220233207469436, −9.050295050096330378420598046106, −8.719321556270734284802601187058, −7.85580790399628805628886730995, −6.77281753931536744906556435957, −6.10235591955112131744994563437, −4.70106053807655435017338492199, −4.12088363715474953985719255507, −3.09501979328661157282564727988, −1.12658009108641099209021411862, 1.52606600413965038918931065043, 2.20046017863110032847023671119, 3.48883552037523235919850889329, 4.54293138336666945559503498623, 5.65937354170702915448443540274, 6.64086211245420819777783884925, 7.973717420166443724630739853945, 8.405104010515670857195370137879, 9.073112285269474811662224072217, 10.18120670337007146593877302958

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