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Properties

Label 2-850-17.13-c1-0-15
Degree $2$
Conductor $850$
Sign $0.963 - 0.266i$
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 + (2.31 + 2.31i)3-s − 4-s + (2.31 − 2.31i)6-s + (3.26 − 3.26i)7-s + i·8-s + 7.68i·9-s + (−1.82 + 1.82i)11-s + (−2.31 − 2.31i)12-s + 0.145·13-s + (−3.26 − 3.26i)14-s + 16-s + (3.31 + 2.45i)17-s + 7.68·18-s − 2.06i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (1.33 + 1.33i)3-s − 0.5·4-s + (0.943 − 0.943i)6-s + (1.23 − 1.23i)7-s + 0.353i·8-s + 2.56i·9-s + (−0.551 + 0.551i)11-s + (−0.667 − 0.667i)12-s + 0.0404·13-s + (−0.873 − 0.873i)14-s + 0.250·16-s + (0.803 + 0.595i)17-s + 1.81·18-s − 0.472i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.963 - 0.266i$
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: \((2,\ 850,\ (\ :1/2),\ 0.963 - 0.266i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.54635 + 0.345233i\)
\(L(\frac12)\) \(\approx\) \(2.54635 + 0.345233i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 - 2.45i)T \)
good3 \( 1 + (-2.31 - 2.31i)T + 3iT^{2} \)
7 \( 1 + (-3.26 + 3.26i)T - 7iT^{2} \)
11 \( 1 + (1.82 - 1.82i)T - 11iT^{2} \)
13 \( 1 - 0.145T + 13T^{2} \)
19 \( 1 + 2.06iT - 19T^{2} \)
23 \( 1 + (-3.41 + 3.41i)T - 23iT^{2} \)
29 \( 1 + (-2.10 - 2.10i)T + 29iT^{2} \)
31 \( 1 + (2.78 + 2.78i)T + 31iT^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 + (5.53 - 5.53i)T - 41iT^{2} \)
43 \( 1 + 0.622iT - 43T^{2} \)
47 \( 1 + 8.47T + 47T^{2} \)
53 \( 1 + 6.68iT - 53T^{2} \)
59 \( 1 - 5.71iT - 59T^{2} \)
61 \( 1 + (-2.63 + 2.63i)T - 61iT^{2} \)
67 \( 1 + 1.58T + 67T^{2} \)
71 \( 1 + (-1.83 - 1.83i)T + 71iT^{2} \)
73 \( 1 + (-5.82 - 5.82i)T + 73iT^{2} \)
79 \( 1 + (-3.29 + 3.29i)T - 79iT^{2} \)
83 \( 1 + 8.91iT - 83T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 + (12.5 + 12.5i)T + 97iT^{2} \)
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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

−10.15976179752424282608791789004, −9.677802728409923846196669978714, −8.485439407081151665654407286568, −8.126907817065276586493589386143, −7.21662609269066891607117631893, −5.10505174051850921781931328043, −4.60803122990648269219789936444, −3.81117219140134024895014239570, −2.85681274026975509876065877358, −1.64082146433733557784162174913, 1.34157374814247450900026558976, 2.47034490802670404638749481459, 3.45120159183274146684320412711, 5.11777002247425183365249151999, 5.87466675140136841831611926738, 6.98238946568047282764662768809, 7.82829240219325790931736557960, 8.224638802041945354705203512479, 8.887940120268604737110526368303, 9.632777179800772761198634076360

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