↑

Properties

Label 2-928-29.21-c0-0-0
Degree $2$
Conductor $928$
Sign $0.487 - 0.873i$
Analytic cond. $0.463132$
Root an. cond. $0.680538$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.846 + 1.75i)5-s + (−0.974 − 0.222i)9-s + (1.21 − 0.277i)13-s + (−0.467 + 0.467i)17-s + (−1.74 + 2.19i)25-s + (−0.974 + 0.222i)29-s + (1.68 − 1.05i)37-s + (−0.158 − 0.158i)41-s + (−0.433 − 1.90i)45-s + (0.222 − 0.974i)49-s + (−0.781 + 0.376i)53-s + (1.87 + 0.211i)61-s + (1.51 + 1.90i)65-s + (−0.623 − 1.78i)73-s + (0.900 + 0.433i)81-s + ⋯
L(s)  = 1  + (0.846 + 1.75i)5-s + (−0.974 − 0.222i)9-s + (1.21 − 0.277i)13-s + (−0.467 + 0.467i)17-s + (−1.74 + 2.19i)25-s + (−0.974 + 0.222i)29-s + (1.68 − 1.05i)37-s + (−0.158 − 0.158i)41-s + (−0.433 − 1.90i)45-s + (0.222 − 0.974i)49-s + (−0.781 + 0.376i)53-s + (1.87 + 0.211i)61-s + (1.51 + 1.90i)65-s + (−0.623 − 1.78i)73-s + (0.900 + 0.433i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(928\)    =    \(2^{5} \cdot 29\)
Sign: $0.487 - 0.873i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{928} (833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 928,\ (\ :0),\ 0.487 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.091705210\)
\(L(\frac12)\) \(\approx\) \(1.091705210\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (0.974 - 0.222i)T \)
good3 \( 1 + (0.974 + 0.222i)T^{2} \)
5 \( 1 + (-0.846 - 1.75i)T + (-0.623 + 0.781i)T^{2} \)
7 \( 1 + (-0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.433 + 0.900i)T^{2} \)
13 \( 1 + (-1.21 + 0.277i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (0.467 - 0.467i)T - iT^{2} \)
19 \( 1 + (-0.974 + 0.222i)T^{2} \)
23 \( 1 + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.781 - 0.623i)T^{2} \)
37 \( 1 + (-1.68 + 1.05i)T + (0.433 - 0.900i)T^{2} \)
41 \( 1 + (0.158 + 0.158i)T + iT^{2} \)
43 \( 1 + (0.781 - 0.623i)T^{2} \)
47 \( 1 + (-0.433 - 0.900i)T^{2} \)
53 \( 1 + (0.781 - 0.376i)T + (0.623 - 0.781i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-1.87 - 0.211i)T + (0.974 + 0.222i)T^{2} \)
67 \( 1 + (0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.623 + 1.78i)T + (-0.781 + 0.623i)T^{2} \)
79 \( 1 + (-0.433 + 0.900i)T^{2} \)
83 \( 1 + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.559 + 1.59i)T + (-0.781 - 0.623i)T^{2} \)
97 \( 1 + (1.97 - 0.222i)T + (0.974 - 0.222i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.61434751136575621531592638984, −9.648565295487022330584666136006, −8.850622856478770288199633279811, −7.82394688944776467652884424874, −6.84165730739571425838368140689, −6.07864482460356253272143398860, −5.63442687226350527913778288997, −3.82786386909435779796869105778, −3.02535038996027162754414776790, −2.03302391219050904038256123295, 1.18672375782254898786166719241, 2.44129805991302587478381299853, 4.01334883377008655683312628203, 4.97516847316289865906735086964, 5.72124758511507812802887384740, 6.41031375069102620664831790076, 7.979896886397177014378245475234, 8.560806845031130285271335805028, 9.216984811974811928240770285218, 9.846781758125136840473988378341

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