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Properties

Label 2-833-17.9-c1-0-30
Degree $2$
Conductor $833$
Sign $-0.00394 - 0.999i$
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  + (−0.574 + 0.574i)2-s + (2.83 + 1.17i)3-s + 1.34i·4-s + (0.724 − 1.74i)5-s + (−2.29 + 0.952i)6-s + (−1.91 − 1.91i)8-s + (4.51 + 4.51i)9-s + (0.588 + 1.42i)10-s + (2.75 − 1.14i)11-s + (−1.57 + 3.79i)12-s + 5.10i·13-s + (4.10 − 4.10i)15-s − 0.475·16-s + (−4.11 − 0.175i)17-s − 5.18·18-s + (0.882 − 0.882i)19-s + ⋯
L(s)  = 1  + (−0.406 + 0.406i)2-s + (1.63 + 0.676i)3-s + 0.670i·4-s + (0.323 − 0.782i)5-s + (−0.938 + 0.388i)6-s + (−0.678 − 0.678i)8-s + (1.50 + 1.50i)9-s + (0.186 + 0.449i)10-s + (0.830 − 0.344i)11-s + (−0.453 + 1.09i)12-s + 1.41i·13-s + (1.05 − 1.05i)15-s − 0.118·16-s + (−0.999 − 0.0425i)17-s − 1.22·18-s + (0.202 − 0.202i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00394 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00394 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.00394 - 0.999i$
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: \((2,\ 833,\ (\ :1/2),\ -0.00394 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62382 + 1.63024i\)
\(L(\frac12)\) \(\approx\) \(1.62382 + 1.63024i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.11 + 0.175i)T \)
good2 \( 1 + (0.574 - 0.574i)T - 2iT^{2} \)
3 \( 1 + (-2.83 - 1.17i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.724 + 1.74i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (-2.75 + 1.14i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 5.10iT - 13T^{2} \)
19 \( 1 + (-0.882 + 0.882i)T - 19iT^{2} \)
23 \( 1 + (2.22 - 0.921i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.0259 + 0.0627i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-5.07 - 2.10i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-2.50 - 1.03i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.92 + 7.06i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-5.69 - 5.69i)T + 43iT^{2} \)
47 \( 1 - 0.811iT - 47T^{2} \)
53 \( 1 + (-3.69 + 3.69i)T - 53iT^{2} \)
59 \( 1 + (6.18 + 6.18i)T + 59iT^{2} \)
61 \( 1 + (3.49 + 8.43i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 3.69T + 67T^{2} \)
71 \( 1 + (6.49 + 2.69i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-5.84 + 14.1i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (15.5 - 6.42i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (2.20 - 2.20i)T - 83iT^{2} \)
89 \( 1 + 1.87iT - 89T^{2} \)
97 \( 1 + (-2.50 + 6.03i)T + (-68.5 - 68.5i)T^{2} \)
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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

−9.811540671385899087823102379336, −9.173337135258774101240691865010, −8.891162553571615398686404353119, −8.228017381444092865188790277341, −7.25317662305816055974743006733, −6.38501187177810291694586617253, −4.63099856286032259648221968155, −4.06102246147629102642856176982, −3.04411056423476130216769293044, −1.82482856119402449360700575643, 1.20831929076911398150387240723, 2.38814074185287646733610475847, 2.92721849036363120558871446130, 4.25912926992712982285039433894, 5.91303272362326639272118541208, 6.69035876168671816811240180582, 7.58100015089435334576947088418, 8.506942120097072769361876509122, 9.102791808156170278227946631720, 10.00208808918333128923464063722

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