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Properties

Label 2-850-85.37-c1-0-8
Degree $2$
Conductor $850$
Sign $-0.784 - 0.620i$
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 + (−2.33 + 1.56i)3-s + (0.707 + 0.707i)4-s + (−2.75 + 0.548i)6-s + (1.27 − 0.254i)7-s + (0.382 + 0.923i)8-s + (1.88 − 4.54i)9-s + (1.14 + 5.73i)11-s + (−2.75 − 0.548i)12-s + 5.33·13-s + (1.27 + 0.254i)14-s + i·16-s + (−3.93 − 1.22i)17-s + (3.47 − 3.47i)18-s + (0.311 + 0.752i)19-s + ⋯
L(s)  = 1  + (0.653 + 0.270i)2-s + (−1.35 + 0.902i)3-s + (0.353 + 0.353i)4-s + (−1.12 + 0.224i)6-s + (0.483 − 0.0961i)7-s + (0.135 + 0.326i)8-s + (0.626 − 1.51i)9-s + (0.343 + 1.72i)11-s + (−0.796 − 0.158i)12-s + 1.48·13-s + (0.341 + 0.0679i)14-s + 0.250i·16-s + (−0.954 − 0.296i)17-s + (0.819 − 0.819i)18-s + (0.0715 + 0.172i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.784 - 0.620i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ -0.784 - 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461107 + 1.32712i\)
\(L(\frac12)\) \(\approx\) \(0.461107 + 1.32712i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.93 + 1.22i)T \)
good3 \( 1 + (2.33 - 1.56i)T + (1.14 - 2.77i)T^{2} \)
7 \( 1 + (-1.27 + 0.254i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (-1.14 - 5.73i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 - 5.33T + 13T^{2} \)
19 \( 1 + (-0.311 - 0.752i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.88 - 5.82i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (-1.17 - 1.75i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (-0.301 + 1.51i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (3.87 + 5.79i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (4.40 - 6.59i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (-1.31 + 0.544i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 0.109iT - 47T^{2} \)
53 \( 1 + (0.379 - 0.915i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-5.08 - 2.10i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.82 + 3.22i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (4.12 - 4.12i)T - 67iT^{2} \)
71 \( 1 + (5.38 + 1.07i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-8.02 - 1.59i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (-5.01 + 0.997i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (-2.38 - 0.987i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (0.779 + 0.779i)T + 89iT^{2} \)
97 \( 1 + (-14.7 - 2.93i)T + (89.6 + 37.1i)T^{2} \)
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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

−10.69143747156640147153696243880, −9.894076743874375920395858728091, −9.008469457385841061216875078742, −7.72770093937071781498848972680, −6.74087453480085347421114530207, −6.02949955301165835519596020892, −5.10589250391248193451858079883, −4.42488698370144637345132545777, −3.71886305511748481650995075248, −1.72175171731884765266356722469, 0.68420269504690702962921505124, 1.83833411021270619985586838340, 3.45050370207269955436197758248, 4.61428463336351685673100562089, 5.66937640338043905619396161053, 6.25020818706313403379382493123, 6.74587585640424822739366155777, 8.179510762470275935773844215601, 8.753683866009343905987241288389, 10.50040231719546433617831784185

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