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Properties

Label 2-833-119.93-c1-0-14
Degree $2$
Conductor $833$
Sign $0.428 + 0.903i$
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  + (−0.624 − 2.33i)2-s + (−1.07 + 0.141i)3-s + (−3.31 + 1.91i)4-s + (0.607 − 0.465i)5-s + (1 + 2.41i)6-s + (3.12 + 3.12i)8-s + (−1.76 + 0.473i)9-s + (−1.46 − 1.12i)10-s + (−1.59 + 2.07i)11-s + (3.28 − 2.52i)12-s + 1.41i·13-s + (−0.585 + 0.585i)15-s + (1.50 − 2.59i)16-s + (1.18 − 3.94i)17-s + (2.20 + 3.82i)18-s + (0.214 + 0.800i)19-s + ⋯
L(s)  = 1  + (−0.441 − 1.64i)2-s + (−0.619 + 0.0815i)3-s + (−1.65 + 0.957i)4-s + (0.271 − 0.208i)5-s + (0.408 + 0.985i)6-s + (1.10 + 1.10i)8-s + (−0.588 + 0.157i)9-s + (−0.463 − 0.355i)10-s + (−0.479 + 0.625i)11-s + (0.949 − 0.728i)12-s + 0.392i·13-s + (−0.151 + 0.151i)15-s + (0.375 − 0.649i)16-s + (0.287 − 0.957i)17-s + (0.520 + 0.901i)18-s + (0.0491 + 0.183i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.428 + 0.903i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.428 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.624196 - 0.394572i\)
\(L(\frac12)\) \(\approx\) \(0.624196 - 0.394572i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 + 3.94i)T \)
good2 \( 1 + (0.624 + 2.33i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (1.07 - 0.141i)T + (2.89 - 0.776i)T^{2} \)
5 \( 1 + (-0.607 + 0.465i)T + (1.29 - 4.82i)T^{2} \)
11 \( 1 + (1.59 - 2.07i)T + (-2.84 - 10.6i)T^{2} \)
13 \( 1 - 1.41iT - 13T^{2} \)
19 \( 1 + (-0.214 - 0.800i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.73 - 0.623i)T + (22.2 + 5.95i)T^{2} \)
29 \( 1 + (0.292 + 0.121i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-7.77 + 1.02i)T + (29.9 - 8.02i)T^{2} \)
37 \( 1 + (-5.62 - 7.32i)T + (-9.57 + 35.7i)T^{2} \)
41 \( 1 + (-1.12 + 0.464i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.585 + 0.585i)T + 43iT^{2} \)
47 \( 1 + (4.47 + 2.58i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.36 - 0.366i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.55 + 5.79i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.499 + 3.79i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (0.585 + 1.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.07 - 5i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-1.68 - 12.8i)T + (-70.5 + 18.8i)T^{2} \)
79 \( 1 + (4.73 + 0.623i)T + (76.3 + 20.4i)T^{2} \)
83 \( 1 + (-8.24 + 8.24i)T - 83iT^{2} \)
89 \( 1 + (-5.70 - 3.29i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.53 - 3.94i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.08745633811450970239633429444, −9.596791133735279288181586138790, −8.743858222702473702857530458086, −7.80750875756892380277893451077, −6.54414567227284432874802962494, −5.24842557436027885991782057470, −4.60508482637185957018074244828, −3.21834559048962873200246537726, −2.32867992856900863897719868944, −0.962180032811343376292840815162, 0.61467831607134358317994542881, 2.91456067259697952799517102215, 4.56710500612433391687161635612, 5.57884965172839315735632760845, 6.04771349836047470722412086298, 6.74485217103139541501925007988, 7.82992447588179079469901262836, 8.417545651456787161222451045556, 9.215440128930318915521503853153, 10.25644004927665751584088046576

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