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Properties

Label 2-833-119.16-c1-0-55
Degree $2$
Conductor $833$
Sign $-0.0346 - 0.999i$
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  + (1.15 − 2.00i)2-s + (0.115 − 0.0667i)3-s + (−1.68 − 2.92i)4-s + (−2.18 − 1.26i)5-s − 0.309i·6-s − 3.19·8-s + (−1.49 + 2.58i)9-s + (−5.07 + 2.92i)10-s + (−4.29 + 2.48i)11-s + (−0.390 − 0.225i)12-s − 2.86·13-s − 0.337·15-s + (−0.331 + 0.573i)16-s + (−3.19 + 2.60i)17-s + (3.45 + 5.99i)18-s + (2.98 − 5.17i)19-s + ⋯
L(s)  = 1  + (0.820 − 1.42i)2-s + (0.0667 − 0.0385i)3-s + (−0.844 − 1.46i)4-s + (−0.978 − 0.564i)5-s − 0.126i·6-s − 1.13·8-s + (−0.497 + 0.860i)9-s + (−1.60 + 0.926i)10-s + (−1.29 + 0.748i)11-s + (−0.112 − 0.0651i)12-s − 0.793·13-s − 0.0870·15-s + (−0.0828 + 0.143i)16-s + (−0.774 + 0.632i)17-s + (0.815 + 1.41i)18-s + (0.685 − 1.18i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0346 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0346 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.0346 - 0.999i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ -0.0346 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.304477 + 0.315228i\)
\(L(\frac12)\) \(\approx\) \(0.304477 + 0.315228i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.19 - 2.60i)T \)
good2 \( 1 + (-1.15 + 2.00i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.115 + 0.0667i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.18 + 1.26i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.29 - 2.48i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.86T + 13T^{2} \)
19 \( 1 + (-2.98 + 5.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.59 - 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.37iT - 29T^{2} \)
31 \( 1 + (2.33 - 1.35i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.41 + 3.12i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.74iT - 41T^{2} \)
43 \( 1 + 6.21T + 43T^{2} \)
47 \( 1 + (0.788 - 1.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0987 - 0.171i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.06 - 7.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.472 - 0.272i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.358 + 0.620i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + (-4.66 + 2.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12.0 - 6.95i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + (8.48 - 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.73iT - 97T^{2} \)
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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

−9.928765097665671839747476066593, −8.897872461020180742175418615087, −7.903207643878317620976985868819, −7.19000576250155887900002754535, −5.28794098626800608744054381893, −4.94443298387792135733673451021, −4.05530200738932691228740310494, −2.83717653740312175916028321825, −2.05815181510093366959111780404, −0.15197690145134078323907594760, 3.00245931137684906308100495780, 3.64050583450964964075349361215, 4.88132078002752064541744543477, 5.56588816500957629652754692378, 6.62065157487354341303826817708, 7.26779959437147901851550264371, 8.026076113010620730789115202761, 8.658057609869900541223110187208, 9.913863209820265767993395371802, 11.00881798078988051217537918319

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