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Properties

Label 2-850-85.23-c1-0-10
Degree $2$
Conductor $850$
Sign $-0.245 + 0.969i$
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  + (−0.923 + 0.382i)2-s + (−2.12 − 1.42i)3-s + (0.707 − 0.707i)4-s + (2.51 + 0.499i)6-s + (−0.0411 − 0.00818i)7-s + (−0.382 + 0.923i)8-s + (1.36 + 3.29i)9-s + (0.476 − 2.39i)11-s + (−2.51 + 0.499i)12-s + 2.17·13-s + (0.0411 − 0.00818i)14-s − i·16-s + (2.37 − 3.36i)17-s + (−2.51 − 2.51i)18-s + (−1.43 + 3.45i)19-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (−1.22 − 0.821i)3-s + (0.353 − 0.353i)4-s + (1.02 + 0.204i)6-s + (−0.0155 − 0.00309i)7-s + (−0.135 + 0.326i)8-s + (0.454 + 1.09i)9-s + (0.143 − 0.721i)11-s + (−0.725 + 0.144i)12-s + 0.601·13-s + (0.0109 − 0.00218i)14-s − 0.250i·16-s + (0.576 − 0.816i)17-s + (−0.593 − 0.593i)18-s + (−0.328 + 0.792i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.245 + 0.969i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ -0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.374478 - 0.480961i\)
\(L(\frac12)\) \(\approx\) \(0.374478 - 0.480961i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 \)
17 \( 1 + (-2.37 + 3.36i)T \)
good3 \( 1 + (2.12 + 1.42i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (0.0411 + 0.00818i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.476 + 2.39i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 - 2.17T + 13T^{2} \)
19 \( 1 + (1.43 - 3.45i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.96 - 5.93i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (-4.73 + 7.08i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-0.226 - 1.13i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-4.07 + 6.10i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-0.400 - 0.599i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (8.37 + 3.46i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 4.13iT - 47T^{2} \)
53 \( 1 + (2.01 + 4.87i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (8.77 - 3.63i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.87 + 3.92i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (11.0 + 11.0i)T + 67iT^{2} \)
71 \( 1 + (-5.22 + 1.04i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (11.1 - 2.22i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (7.04 + 1.40i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-15.6 + 6.46i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-11.5 + 11.5i)T - 89iT^{2} \)
97 \( 1 + (-4.02 + 0.800i)T + (89.6 - 37.1i)T^{2} \)
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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

−10.02665806406109035978324007402, −9.061107637480024257334979133218, −8.054603876458207170910102201224, −7.34916618509026901483985859975, −6.39780977965483882917652449827, −5.88491664900162869685091017738, −4.99967971810615021778250274597, −3.35041286374311290621614964777, −1.62942479661605859662428200100, −0.51408245678426111726077653120, 1.21569721416978961078343875859, 2.97470488577860631386252467867, 4.31439596250327602947984064266, 4.99014714000913374154211746020, 6.22571634549238424093090007297, 6.77083217780988735493591888149, 8.063829174162432076913170229242, 8.944760870258222702070941667944, 9.812836338240012279957954625128, 10.55911977571246952257652691479

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