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Properties

Label 2-833-17.9-c1-0-41
Degree $2$
Conductor $833$
Sign $0.857 + 0.515i$
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  + (0.751 − 0.751i)2-s + (1.42 + 0.592i)3-s + 0.871i·4-s + (1.27 − 3.08i)5-s + (1.51 − 0.629i)6-s + (2.15 + 2.15i)8-s + (−0.427 − 0.427i)9-s + (−1.35 − 3.28i)10-s + (1.09 − 0.452i)11-s + (−0.515 + 1.24i)12-s + 5.97i·13-s + (3.65 − 3.65i)15-s + 1.49·16-s + (2.28 − 3.43i)17-s − 0.642·18-s + (4.06 − 4.06i)19-s + ⋯
L(s)  = 1  + (0.531 − 0.531i)2-s + (0.825 + 0.341i)3-s + 0.435i·4-s + (0.572 − 1.38i)5-s + (0.620 − 0.256i)6-s + (0.762 + 0.762i)8-s + (−0.142 − 0.142i)9-s + (−0.429 − 1.03i)10-s + (0.329 − 0.136i)11-s + (−0.148 + 0.359i)12-s + 1.65i·13-s + (0.944 − 0.944i)15-s + 0.374·16-s + (0.554 − 0.832i)17-s − 0.151·18-s + (0.932 − 0.932i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.857 + 0.515i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.857 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.92489 - 0.811553i\)
\(L(\frac12)\) \(\approx\) \(2.92489 - 0.811553i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.28 + 3.43i)T \)
good2 \( 1 + (-0.751 + 0.751i)T - 2iT^{2} \)
3 \( 1 + (-1.42 - 0.592i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.27 + 3.08i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (-1.09 + 0.452i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 5.97iT - 13T^{2} \)
19 \( 1 + (-4.06 + 4.06i)T - 19iT^{2} \)
23 \( 1 + (2.85 - 1.18i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.715 - 1.72i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-6.73 - 2.78i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (4.57 + 1.89i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.272 - 0.658i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (7.40 + 7.40i)T + 43iT^{2} \)
47 \( 1 - 8.39iT - 47T^{2} \)
53 \( 1 + (6.01 - 6.01i)T - 53iT^{2} \)
59 \( 1 + (-0.465 - 0.465i)T + 59iT^{2} \)
61 \( 1 + (3.79 + 9.16i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 0.217T + 67T^{2} \)
71 \( 1 + (-10.0 - 4.16i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.83 - 4.43i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.35 + 0.562i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (3.20 - 3.20i)T - 83iT^{2} \)
89 \( 1 - 5.20iT - 89T^{2} \)
97 \( 1 + (-5.46 + 13.1i)T + (-68.5 - 68.5i)T^{2} \)
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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

−9.810777509443957954693553639903, −9.203221732956661274090711197555, −8.743684119388076816093455628895, −7.84432997397648479285345997086, −6.68075075453521084798723983430, −5.27852764297344105984951437053, −4.59754007370007659642947302505, −3.70650135284237022760294340342, −2.67226292216145424584291980985, −1.48433692796199570819269687864, 1.64482058564276216970415490342, 2.88345793327242211911085169145, 3.66214663513836021828869041799, 5.28189454999562926886573195769, 5.97506117461144174997562874419, 6.70377100139870447778963536494, 7.72839987150003333619935354480, 8.182207197979200574801718736955, 9.758367389701644910333554400967, 10.16601433771429536089781456209

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