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Properties

Label 2-833-119.67-c1-0-28
Degree $2$
Conductor $833$
Sign $-0.487 - 0.873i$
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  + (0.974 + 1.68i)2-s + (2.01 + 1.16i)3-s + (−0.900 + 1.55i)4-s + (0.826 − 0.477i)5-s + 4.53i·6-s + 0.389·8-s + (1.20 + 2.08i)9-s + (1.61 + 0.930i)10-s + (−0.599 − 0.346i)11-s + (−3.62 + 2.09i)12-s − 0.863·13-s + 2.22·15-s + (2.17 + 3.77i)16-s + (−3.13 + 2.68i)17-s + (−2.34 + 4.06i)18-s + (0.0357 + 0.0619i)19-s + ⋯
L(s)  = 1  + (0.689 + 1.19i)2-s + (1.16 + 0.671i)3-s + (−0.450 + 0.779i)4-s + (0.369 − 0.213i)5-s + 1.85i·6-s + 0.137·8-s + (0.401 + 0.695i)9-s + (0.509 + 0.294i)10-s + (−0.180 − 0.104i)11-s + (−1.04 + 0.604i)12-s − 0.239·13-s + 0.573·15-s + (0.544 + 0.943i)16-s + (−0.759 + 0.650i)17-s + (−0.553 + 0.959i)18-s + (0.00821 + 0.0142i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $-0.487 - 0.873i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ -0.487 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75009 + 2.97999i\)
\(L(\frac12)\) \(\approx\) \(1.75009 + 2.97999i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.13 - 2.68i)T \)
good2 \( 1 + (-0.974 - 1.68i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-2.01 - 1.16i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.826 + 0.477i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.599 + 0.346i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.863T + 13T^{2} \)
19 \( 1 + (-0.0357 - 0.0619i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.95 + 2.85i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.92iT - 29T^{2} \)
31 \( 1 + (5.70 + 3.29i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.00 + 5.19i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.48iT - 41T^{2} \)
43 \( 1 + 0.410T + 43T^{2} \)
47 \( 1 + (3.80 + 6.58i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.555 - 0.962i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.54 - 6.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.16 - 0.673i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.65 + 6.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.0iT - 71T^{2} \)
73 \( 1 + (-0.487 - 0.281i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.48 - 3.74i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.32T + 83T^{2} \)
89 \( 1 + (7.37 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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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

−10.36493990771438272477990160722, −9.237796982957669488066681857650, −8.814819550481133904020155844666, −7.85240281202114632460172418059, −7.09218749092660246424400164469, −6.05425250991168014169183637695, −5.17184319671607175311318591108, −4.28645845040253983557244199016, −3.42780786617737494035879964630, −2.07045707003193216924068140698, 1.43889649187275396769333183815, 2.52091110913769976402814285286, 2.95356209914647577882284377618, 4.18086366169590302538097237835, 5.16350320275369221952598673565, 6.54173085687391342063238362850, 7.49912725138951153601903382009, 8.204679552444657631507326457724, 9.376152554463592599531250995148, 9.866185011045894481902842047414

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