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Properties

Label 2-23-23.22-c14-0-11
Degree $2$
Conductor $23$
Sign $0.595 - 0.803i$
Analytic cond. $28.5956$
Root an. cond. $5.34749$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 181.·2-s + 3.18e3·3-s + 1.66e4·4-s − 6.72e4i·5-s − 5.79e5·6-s + 3.11e5i·7-s − 5.05e4·8-s + 5.38e6·9-s + 1.22e7i·10-s + 3.75e7i·11-s + 5.31e7·12-s + 2.15e6·13-s − 5.65e7i·14-s − 2.14e8i·15-s − 2.63e8·16-s − 3.10e8i·17-s + ⋯
L(s)  = 1  − 1.42·2-s + 1.45·3-s + 1.01·4-s − 0.860i·5-s − 2.07·6-s + 0.377i·7-s − 0.0240·8-s + 1.12·9-s + 1.22i·10-s + 1.92i·11-s + 1.48·12-s + 0.0343·13-s − 0.536i·14-s − 1.25i·15-s − 0.982·16-s − 0.757i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.595 - 0.803i$
Analytic conductor: \(28.5956\)
Root analytic conductor: \(5.34749\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :7),\ 0.595 - 0.803i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(1.417315700\)
\(L(\frac12)\) \(\approx\) \(1.417315700\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (2.02e9 - 2.73e9i)T \)
good2 \( 1 + 181.T + 1.63e4T^{2} \)
3 \( 1 - 3.18e3T + 4.78e6T^{2} \)
5 \( 1 + 6.72e4iT - 6.10e9T^{2} \)
7 \( 1 - 3.11e5iT - 6.78e11T^{2} \)
11 \( 1 - 3.75e7iT - 3.79e14T^{2} \)
13 \( 1 - 2.15e6T + 3.93e15T^{2} \)
17 \( 1 + 3.10e8iT - 1.68e17T^{2} \)
19 \( 1 - 4.08e8iT - 7.99e17T^{2} \)
29 \( 1 - 4.76e9T + 2.97e20T^{2} \)
31 \( 1 - 3.01e10T + 7.56e20T^{2} \)
37 \( 1 - 1.00e10iT - 9.01e21T^{2} \)
41 \( 1 + 3.94e10T + 3.79e22T^{2} \)
43 \( 1 - 3.58e11iT - 7.38e22T^{2} \)
47 \( 1 + 2.29e11T + 2.56e23T^{2} \)
53 \( 1 - 1.68e12iT - 1.37e24T^{2} \)
59 \( 1 - 2.91e12T + 6.19e24T^{2} \)
61 \( 1 - 5.28e12iT - 9.87e24T^{2} \)
67 \( 1 + 2.07e12iT - 3.67e25T^{2} \)
71 \( 1 + 4.52e12T + 8.27e25T^{2} \)
73 \( 1 - 2.15e13T + 1.22e26T^{2} \)
79 \( 1 + 8.95e12iT - 3.68e26T^{2} \)
83 \( 1 + 1.16e13iT - 7.36e26T^{2} \)
89 \( 1 + 1.38e13iT - 1.95e27T^{2} \)
97 \( 1 - 1.14e14iT - 6.52e27T^{2} \)
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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

−14.97893473226131628382828567418, −13.52907082657555017588764095057, −12.13700609242698109712180417568, −9.932194215018272511547878336326, −9.292498944222907575766205981094, −8.274686165431072021314637492674, −7.31922125926393765891266485536, −4.52889192189304476736310180636, −2.40566013600864462378185788832, −1.32090356086817788680920468489, 0.65400534732543583575420823204, 2.34486215749819563464509148278, 3.56617056627770486914707187209, 6.67849629047638004706072134764, 8.130622429633193445272807068600, 8.682675571434219140829822321009, 10.11401317941878032396146467961, 11.07661429469123832896329945647, 13.54362707719344157237249474985, 14.32165173176799065068781186104

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