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Properties

Label 2-850-17.13-c1-0-6
Degree $2$
Conductor $850$
Sign $-0.319 - 0.947i$
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 + (2.25 + 2.25i)3-s − 4-s + (2.25 − 2.25i)6-s + (−3.11 + 3.11i)7-s + i·8-s + 7.19i·9-s + (0.512 − 0.512i)11-s + (−2.25 − 2.25i)12-s − 5.14·13-s + (3.11 + 3.11i)14-s + 16-s + (2.27 − 3.44i)17-s + 7.19·18-s − 0.658i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (1.30 + 1.30i)3-s − 0.5·4-s + (0.921 − 0.921i)6-s + (−1.17 + 1.17i)7-s + 0.353i·8-s + 2.39i·9-s + (0.154 − 0.154i)11-s + (−0.651 − 0.651i)12-s − 1.42·13-s + (0.831 + 0.831i)14-s + 0.250·16-s + (0.550 − 0.834i)17-s + 1.69·18-s − 0.151i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.319 - 0.947i$
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: \((2,\ 850,\ (\ :1/2),\ -0.319 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917382 + 1.27683i\)
\(L(\frac12)\) \(\approx\) \(0.917382 + 1.27683i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.27 + 3.44i)T \)
good3 \( 1 + (-2.25 - 2.25i)T + 3iT^{2} \)
7 \( 1 + (3.11 - 3.11i)T - 7iT^{2} \)
11 \( 1 + (-0.512 + 0.512i)T - 11iT^{2} \)
13 \( 1 + 5.14T + 13T^{2} \)
19 \( 1 + 0.658iT - 19T^{2} \)
23 \( 1 + (4.04 - 4.04i)T - 23iT^{2} \)
29 \( 1 + (-5.49 - 5.49i)T + 29iT^{2} \)
31 \( 1 + (3.45 + 3.45i)T + 31iT^{2} \)
37 \( 1 + (-3.19 - 3.19i)T + 37iT^{2} \)
41 \( 1 + (-4.66 + 4.66i)T - 41iT^{2} \)
43 \( 1 - 7.03iT - 43T^{2} \)
47 \( 1 - 6.90T + 47T^{2} \)
53 \( 1 - 5.37iT - 53T^{2} \)
59 \( 1 - 0.733iT - 59T^{2} \)
61 \( 1 + (-3.36 + 3.36i)T - 61iT^{2} \)
67 \( 1 - 2.59T + 67T^{2} \)
71 \( 1 + (-8.08 - 8.08i)T + 71iT^{2} \)
73 \( 1 + (-1.14 - 1.14i)T + 73iT^{2} \)
79 \( 1 + (6.60 - 6.60i)T - 79iT^{2} \)
83 \( 1 - 5.80iT - 83T^{2} \)
89 \( 1 + 1.55T + 89T^{2} \)
97 \( 1 + (-10.8 - 10.8i)T + 97iT^{2} \)
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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

−9.926216282402000343629233785232, −9.686640075070022299035830542839, −9.154619305589726951943072858739, −8.318175454191500816553915733529, −7.30160980462736757934155280415, −5.69541399009357954139968505678, −4.86486547192295915380974681527, −3.79027336187469461714953060643, −2.87617242857728722239889325347, −2.43092633167244099798328100051, 0.62638637715450849868468609906, 2.26980223680308489461660703601, 3.41059237099961951800388600461, 4.28117832781187501697055844243, 6.06012955406267633933278453816, 6.74329883940774772355473276655, 7.41915861715607385801363220843, 7.925346034656174256947962116510, 8.876131570953618686272496523050, 9.782660848457036330279463324006

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