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Properties

Label 2-850-25.6-c1-0-31
Degree $2$
Conductor $850$
Sign $-0.535 + 0.844i$
Analytic cond. $6.78728$
Root an. cond. $2.60524$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.690 + 2.12i)5-s + (0.809 + 0.587i)6-s − 4.23·7-s + (−0.809 − 0.587i)8-s + (−0.618 + 1.90i)9-s − 2.23·10-s + (−0.690 − 2.12i)11-s + (−0.309 + 0.951i)12-s + (1.11 − 3.44i)13-s + (−1.30 − 4.02i)14-s + (0.690 + 2.12i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.309 + 0.951i)5-s + (0.330 + 0.239i)6-s − 1.60·7-s + (−0.286 − 0.207i)8-s + (−0.206 + 0.634i)9-s − 0.707·10-s + (−0.208 − 0.641i)11-s + (−0.0892 + 0.274i)12-s + (0.310 − 0.954i)13-s + (−0.349 − 1.07i)14-s + (0.178 + 0.549i)15-s + (0.0772 − 0.237i)16-s + (−0.196 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.535 + 0.844i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ -0.535 + 0.844i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.690 - 2.12i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-0.809 + 0.587i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 4.23T + 7T^{2} \)
11 \( 1 + (0.690 + 2.12i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.11 + 3.44i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (3.92 + 2.85i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1 + 3.07i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-6.85 + 4.97i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.80 + 2.04i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.64 - 8.14i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.118 + 0.363i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 + (8.23 - 5.98i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.42 - 4.66i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.52 - 7.77i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.52 - 10.8i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.92 + 5.03i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (4.66 - 3.38i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.33 - 4.11i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.16 - 5.20i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.85 + 4.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.69 + 14.4i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.0729 - 0.0530i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.989350998314240136847261681442, −8.729696324066883733537174381384, −8.150167611183961272576543143827, −7.23206089847342829498893566560, −6.44043191033968806180000951387, −5.90324903496849661326969241033, −4.44136706128099674441406538929, −3.10062680023013353181597163516, −2.79770283917507351596229168991, 0, 1.83323765291067446683071704496, 3.32849811484323287906651288853, 3.86125534458329329862216993767, 4.85895144864186151110376487457, 6.11046419091086427462910417422, 6.88641869148115212359990669625, 8.355996566862826321683247837621, 9.010298521244291727529496679585, 9.618487625030307645095967127680

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