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Properties

Label 2-850-17.13-c1-0-4
Degree $2$
Conductor $850$
Sign $0.122 - 0.992i$
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 + (−1.70 − 1.70i)3-s − 4-s + (1.70 − 1.70i)6-s + (1 − i)7-s − i·8-s + 2.82i·9-s + (−4.41 + 4.41i)11-s + (1.70 + 1.70i)12-s − 3·13-s + (1 + i)14-s + 16-s + (3.53 − 2.12i)17-s − 2.82·18-s + 1.24i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.985 − 0.985i)3-s − 0.5·4-s + (0.696 − 0.696i)6-s + (0.377 − 0.377i)7-s − 0.353i·8-s + 0.942i·9-s + (−1.33 + 1.33i)11-s + (0.492 + 0.492i)12-s − 0.832·13-s + (0.267 + 0.267i)14-s + 0.250·16-s + (0.857 − 0.514i)17-s − 0.666·18-s + 0.285i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.122 - 0.992i$
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: \((2,\ 850,\ (\ :1/2),\ 0.122 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522479 + 0.462103i\)
\(L(\frac12)\) \(\approx\) \(0.522479 + 0.462103i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.53 + 2.12i)T \)
good3 \( 1 + (1.70 + 1.70i)T + 3iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 + (4.41 - 4.41i)T - 11iT^{2} \)
13 \( 1 + 3T + 13T^{2} \)
19 \( 1 - 1.24iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T + 29iT^{2} \)
31 \( 1 + (-7.36 - 7.36i)T + 31iT^{2} \)
37 \( 1 + (-3.24 - 3.24i)T + 37iT^{2} \)
41 \( 1 + (-1.58 + 1.58i)T - 41iT^{2} \)
43 \( 1 - 12.2iT - 43T^{2} \)
47 \( 1 + 4.41T + 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 - 6.89iT - 59T^{2} \)
61 \( 1 + (-1.87 + 1.87i)T - 61iT^{2} \)
67 \( 1 + 2.48T + 67T^{2} \)
71 \( 1 + (-2.29 - 2.29i)T + 71iT^{2} \)
73 \( 1 + (-4.36 - 4.36i)T + 73iT^{2} \)
79 \( 1 + (8.24 - 8.24i)T - 79iT^{2} \)
83 \( 1 + 4.24iT - 83T^{2} \)
89 \( 1 - 5.48T + 89T^{2} \)
97 \( 1 + (-4.12 - 4.12i)T + 97iT^{2} \)
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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

−10.27624474015940177329176800362, −9.718914371207708778886746517508, −8.238805185261030409048724304679, −7.53003385816298857547270713654, −7.07152287038745670880927582429, −6.17449300510913451852019273423, −5.04742784536794516547309249578, −4.73253155014216476074093053538, −2.71768252915778732549344941320, −1.15265858490919283825228503540, 0.44846193960612983258627415134, 2.46631593830180160381168988013, 3.56302134106604976887802193564, 4.74565234729997758223112647398, 5.38493266464052538643794765076, 6.00393878774375152931276734113, 7.66790671866932202662061655760, 8.430958584622497893132207773772, 9.518600852887642611838656241235, 10.22051627843030231144821281812

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