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Properties

Label 2-850-17.13-c1-0-5
Degree $2$
Conductor $850$
Sign $0.615 + 0.788i$
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 + (−2.15 − 2.15i)3-s − 4-s + (−2.15 + 2.15i)6-s + (−2.15 + 2.15i)7-s + i·8-s + 6.31i·9-s + (1 − i)11-s + (2.15 + 2.15i)12-s + 13-s + (2.15 + 2.15i)14-s + 16-s + (1 − 4i)17-s + 6.31·18-s + 8.31i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.24 − 1.24i)3-s − 0.5·4-s + (−0.881 + 0.881i)6-s + (−0.815 + 0.815i)7-s + 0.353i·8-s + 2.10i·9-s + (0.301 − 0.301i)11-s + (0.623 + 0.623i)12-s + 0.277·13-s + (0.576 + 0.576i)14-s + 0.250·16-s + (0.242 − 0.970i)17-s + 1.48·18-s + 1.90i·19-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.651282 - 0.317779i\)
\(L(\frac12)\) \(\approx\) \(0.651282 - 0.317779i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 4i)T \)
good3 \( 1 + (2.15 + 2.15i)T + 3iT^{2} \)
7 \( 1 + (2.15 - 2.15i)T - 7iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 - T + 13T^{2} \)
19 \( 1 - 8.31iT - 19T^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 + (2.31 + 2.31i)T + 29iT^{2} \)
31 \( 1 + (-3.15 - 3.15i)T + 31iT^{2} \)
37 \( 1 + (-6.31 - 6.31i)T + 37iT^{2} \)
41 \( 1 + (-5.31 + 5.31i)T - 41iT^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 11.9iT - 53T^{2} \)
59 \( 1 + 4.63iT - 59T^{2} \)
61 \( 1 + (-5.31 + 5.31i)T - 61iT^{2} \)
67 \( 1 + 6.63T + 67T^{2} \)
71 \( 1 + (6.15 + 6.15i)T + 71iT^{2} \)
73 \( 1 + (-8.63 - 8.63i)T + 73iT^{2} \)
79 \( 1 + (-4.47 + 4.47i)T - 79iT^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 - 18.6T + 89T^{2} \)
97 \( 1 + (-2 - 2i)T + 97iT^{2} \)
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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

−10.24369083525245923316796929465, −9.402367048632181107102742493863, −8.309848078418154724476909155974, −7.44033166284961412927115363477, −6.27173378500449047404268514757, −5.93393451558622080991847783171, −4.91405702872175874499322482777, −3.38635555081881603924759366896, −2.15283684173537381639826381212, −0.921258834982880779423658928219, 0.58361640129880188742599042866, 3.39429883558116195054339198829, 4.27770227063115703048012609911, 4.93529705532713038212415927376, 6.06798479523632449368685299690, 6.53580052835399288230122267044, 7.50255141866587921119106829767, 8.891837079704724511359799894465, 9.623232213751646464714979058747, 10.17215779723071837857357410974

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