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Properties

Label 2-833-17.4-c1-0-40
Degree $2$
Conductor $833$
Sign $-0.615 + 0.788i$
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  − i·2-s + (−2.15 + 2.15i)3-s + 4-s + (1 − i)5-s + (2.15 + 2.15i)6-s − 3i·8-s − 6.31i·9-s + (−1 − i)10-s + (−0.841 − 0.841i)11-s + (−2.15 + 2.15i)12-s − 3.31·13-s + 4.31i·15-s − 16-s + (−4 + i)17-s − 6.31·18-s − 4.31i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.24 + 1.24i)3-s + 0.5·4-s + (0.447 − 0.447i)5-s + (0.881 + 0.881i)6-s − 1.06i·8-s − 2.10i·9-s + (−0.316 − 0.316i)10-s + (−0.253 − 0.253i)11-s + (−0.623 + 0.623i)12-s − 0.919·13-s + 1.11i·15-s − 0.250·16-s + (−0.970 + 0.242i)17-s − 1.48·18-s − 0.990i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ -0.615 + 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.307888 - 0.631010i\)
\(L(\frac12)\) \(\approx\) \(0.307888 - 0.631010i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - i)T \)
good2 \( 1 + iT - 2T^{2} \)
3 \( 1 + (2.15 - 2.15i)T - 3iT^{2} \)
5 \( 1 + (-1 + i)T - 5iT^{2} \)
11 \( 1 + (0.841 + 0.841i)T + 11iT^{2} \)
13 \( 1 + 3.31T + 13T^{2} \)
19 \( 1 + 4.31iT - 19T^{2} \)
23 \( 1 + (3 + 3i)T + 23iT^{2} \)
29 \( 1 + (1.31 - 1.31i)T - 29iT^{2} \)
31 \( 1 + (-5.31 + 5.31i)T - 31iT^{2} \)
37 \( 1 + (6.63 - 6.63i)T - 37iT^{2} \)
41 \( 1 + (1.31 + 1.31i)T + 41iT^{2} \)
43 \( 1 + 8.63iT - 43T^{2} \)
47 \( 1 + 4.31T + 47T^{2} \)
53 \( 1 + 9.63iT - 53T^{2} \)
59 \( 1 + 10.6iT - 59T^{2} \)
61 \( 1 + (-1.68 - 1.68i)T + 61iT^{2} \)
67 \( 1 - 2.31T + 67T^{2} \)
71 \( 1 + (-0.158 + 0.158i)T - 71iT^{2} \)
73 \( 1 + (-5 + 5i)T - 73iT^{2} \)
79 \( 1 + (-5.47 - 5.47i)T + 79iT^{2} \)
83 \( 1 + 4.63iT - 83T^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 + (8.94 - 8.94i)T - 97iT^{2} \)
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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

−9.996344828305264310965503697052, −9.636579754419544923427789056226, −8.567528683700019651818353576604, −6.97787678640604418719072564106, −6.28434354537388803991964887358, −5.24863154785204799578442155059, −4.59926697628781139572114520059, −3.51212859876705616562244671190, −2.16487272795190921765323480040, −0.36859198158612203429157943122, 1.71150544167818824438674631727, 2.57342262619309884353217740176, 4.75099409975859832717689863662, 5.68372386862732304180403548106, 6.25521292052051616837337738225, 6.98614607763300866586562908954, 7.51755155722506997955057198025, 8.364144769578024915575667409846, 9.920335484130708213913254335799, 10.67110241689962704832483635272

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