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Properties

Label 2-833-17.8-c1-0-36
Degree $2$
Conductor $833$
Sign $0.945 - 0.325i$
Analytic cond. $6.65153$
Root an. cond. $2.57905$
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  + (−1.12 + 1.12i)2-s + (−1.18 + 2.86i)3-s − 0.546i·4-s + (0.640 + 0.265i)5-s + (−1.89 − 4.56i)6-s + (−1.64 − 1.64i)8-s + (−4.66 − 4.66i)9-s + (−1.02 + 0.423i)10-s + (−0.0234 − 0.0566i)11-s + (1.56 + 0.647i)12-s − 3.53i·13-s + (−1.52 + 1.52i)15-s + 4.79·16-s + (3.55 + 2.08i)17-s + 10.5·18-s + (1.57 − 1.57i)19-s + ⋯
L(s)  = 1  + (−0.797 + 0.797i)2-s + (−0.684 + 1.65i)3-s − 0.273i·4-s + (0.286 + 0.118i)5-s + (−0.772 − 1.86i)6-s + (−0.579 − 0.579i)8-s + (−1.55 − 1.55i)9-s + (−0.323 + 0.133i)10-s + (−0.00707 − 0.0170i)11-s + (0.451 + 0.187i)12-s − 0.979i·13-s + (−0.392 + 0.392i)15-s + 1.19·16-s + (0.862 + 0.505i)17-s + 2.48·18-s + (0.361 − 0.361i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.945 - 0.325i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.945 - 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367837 + 0.0615052i\)
\(L(\frac12)\) \(\approx\) \(0.367837 + 0.0615052i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 + (-3.55 - 2.08i)T \)
good2 \( 1 + (1.12 - 1.12i)T - 2iT^{2} \)
3 \( 1 + (1.18 - 2.86i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.640 - 0.265i)T + (3.53 + 3.53i)T^{2} \)
11 \( 1 + (0.0234 + 0.0566i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 3.53iT - 13T^{2} \)
19 \( 1 + (-1.57 + 1.57i)T - 19iT^{2} \)
23 \( 1 + (3.23 + 7.81i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.33 - 0.554i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.40 + 5.81i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (1.47 - 3.56i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (8.19 - 3.39i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (7.62 + 7.62i)T + 43iT^{2} \)
47 \( 1 + 8.72iT - 47T^{2} \)
53 \( 1 + (8.97 - 8.97i)T - 53iT^{2} \)
59 \( 1 + (-2.59 - 2.59i)T + 59iT^{2} \)
61 \( 1 + (-7.28 + 3.01i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 8.23T + 67T^{2} \)
71 \( 1 + (-3.63 + 8.77i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.49 + 2.27i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (1.48 + 3.59i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (2.26 - 2.26i)T - 83iT^{2} \)
89 \( 1 - 17.0iT - 89T^{2} \)
97 \( 1 + (2.07 + 0.860i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.19413074922498084228900829326, −9.577220505885670107011887943399, −8.547821212151183047823548832466, −8.018341340944946049329505379112, −6.61149876862561226835825850954, −5.93796039285201865711988955750, −5.10799394864723161286892523362, −3.98563780004592055555095187895, −3.03817842590646229810947895648, −0.27642379562136387563109429052, 1.34472250672949776970663084842, 1.81197847720354704257861157535, 3.18444116537690459115643480940, 5.22089973824500316061665974165, 5.86341416710841220338213824382, 6.82328070059228956140719316612, 7.67461281672450032909591585882, 8.428254232957498293736529781452, 9.505527523177394966083716436197, 10.10987017123283714134195518990

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