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Properties

Label 2-833-119.48-c1-0-20
Degree $2$
Conductor $833$
Sign $0.954 - 0.298i$
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  + (−2.14 + 0.888i)2-s + (−0.289 − 0.0576i)3-s + (2.40 − 2.40i)4-s + (0.677 + 1.01i)5-s + (0.673 − 0.133i)6-s + (−1.24 + 2.99i)8-s + (−2.69 − 1.11i)9-s + (−2.35 − 1.57i)10-s + (2.84 − 0.565i)11-s + (−0.834 + 0.557i)12-s + (−0.992 + 0.992i)13-s + (−0.138 − 0.333i)15-s − 0.737i·16-s + (1.27 − 3.92i)17-s + 6.76·18-s + (1.25 + 3.02i)19-s + ⋯
L(s)  = 1  + (−1.51 + 0.628i)2-s + (−0.167 − 0.0332i)3-s + (1.20 − 1.20i)4-s + (0.303 + 0.453i)5-s + (0.274 − 0.0546i)6-s + (−0.438 + 1.05i)8-s + (−0.896 − 0.371i)9-s + (−0.745 − 0.497i)10-s + (0.856 − 0.170i)11-s + (−0.240 + 0.160i)12-s + (−0.275 + 0.275i)13-s + (−0.0356 − 0.0860i)15-s − 0.184i·16-s + (0.308 − 0.951i)17-s + 1.59·18-s + (0.287 + 0.694i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.954 - 0.298i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.954 - 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652952 + 0.0996561i\)
\(L(\frac12)\) \(\approx\) \(0.652952 + 0.0996561i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 + 3.92i)T \)
good2 \( 1 + (2.14 - 0.888i)T + (1.41 - 1.41i)T^{2} \)
3 \( 1 + (0.289 + 0.0576i)T + (2.77 + 1.14i)T^{2} \)
5 \( 1 + (-0.677 - 1.01i)T + (-1.91 + 4.61i)T^{2} \)
11 \( 1 + (-2.84 + 0.565i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (0.992 - 0.992i)T - 13iT^{2} \)
19 \( 1 + (-1.25 - 3.02i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.353 - 1.77i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (1.94 + 2.90i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (0.338 - 1.70i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-0.119 + 0.603i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-4.22 + 6.31i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (2.34 + 0.969i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (-8.15 + 8.15i)T - 47iT^{2} \)
53 \( 1 + (7.36 - 3.05i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-7.82 - 3.24i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-8.05 - 5.37i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + 4.17iT - 67T^{2} \)
71 \( 1 + (-3.14 + 15.8i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (-7.64 - 11.4i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-11.7 + 2.32i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (7.64 - 3.16i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-6.08 - 6.08i)T + 89iT^{2} \)
97 \( 1 + (3.34 - 2.23i)T + (37.1 - 89.6i)T^{2} \)
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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

−9.979346631253842381893045255257, −9.292570402021926291794762078154, −8.707529421603585888266400390950, −7.73443183769389289204856753212, −6.92887840524880334429374127064, −6.24135399751136271575153473456, −5.41010581196653235040562669744, −3.69483071416883904974319893210, −2.27135927009232604090544602518, −0.73902780984679227305838165247, 0.959169238840919491636819554037, 2.14697932540976122272001797434, 3.30535586025653006054741087401, 4.84492019236232536341238384095, 5.90640412189236570767561560627, 7.02416798259618153846069674817, 7.999877791185735511240099331556, 8.674247494270873922565566090024, 9.353499537367482310433151345725, 9.996335940840335837327111025625

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