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Properties

Label 2-24e2-16.13-c1-0-8
Degree $2$
Conductor $576$
Sign $-0.0313 + 0.999i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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  + (2.37 − 2.37i)5-s − 3.64i·7-s + (−0.841 + 0.841i)11-s + (−2.64 − 2.64i)13-s − 3.06·17-s + (−1.64 − 1.64i)19-s + 7.82i·23-s − 6.29i·25-s + (0.692 + 0.692i)29-s + 0.354·31-s + (−8.66 − 8.66i)35-s + (4.64 − 4.64i)37-s + 6.43i·41-s + (5.64 − 5.64i)43-s + 11.1·47-s + ⋯
L(s)  = 1  + (1.06 − 1.06i)5-s − 1.37i·7-s + (−0.253 + 0.253i)11-s + (−0.733 − 0.733i)13-s − 0.744·17-s + (−0.377 − 0.377i)19-s + 1.63i·23-s − 1.25i·25-s + (0.128 + 0.128i)29-s + 0.0636·31-s + (−1.46 − 1.46i)35-s + (0.763 − 0.763i)37-s + 1.00i·41-s + (0.860 − 0.860i)43-s + 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0313 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.0313 + 0.999i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.0313 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07074 - 1.10483i\)
\(L(\frac12)\) \(\approx\) \(1.07074 - 1.10483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-2.37 + 2.37i)T - 5iT^{2} \)
7 \( 1 + 3.64iT - 7T^{2} \)
11 \( 1 + (0.841 - 0.841i)T - 11iT^{2} \)
13 \( 1 + (2.64 + 2.64i)T + 13iT^{2} \)
17 \( 1 + 3.06T + 17T^{2} \)
19 \( 1 + (1.64 + 1.64i)T + 19iT^{2} \)
23 \( 1 - 7.82iT - 23T^{2} \)
29 \( 1 + (-0.692 - 0.692i)T + 29iT^{2} \)
31 \( 1 - 0.354T + 31T^{2} \)
37 \( 1 + (-4.64 + 4.64i)T - 37iT^{2} \)
41 \( 1 - 6.43iT - 41T^{2} \)
43 \( 1 + (-5.64 + 5.64i)T - 43iT^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + (-5.44 + 5.44i)T - 53iT^{2} \)
59 \( 1 + (-7.82 + 7.82i)T - 59iT^{2} \)
61 \( 1 + (-4.64 - 4.64i)T + 61iT^{2} \)
67 \( 1 + (4 + 4i)T + 67iT^{2} \)
71 \( 1 + 3.36iT - 71T^{2} \)
73 \( 1 - 7.29iT - 73T^{2} \)
79 \( 1 + 4.35T + 79T^{2} \)
83 \( 1 + (-0.841 - 0.841i)T + 83iT^{2} \)
89 \( 1 - 9.50iT - 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−10.32224243604197026881078405783, −9.698156266332769518709248244318, −8.903032458710396578465772600015, −7.76279794067195601147150617532, −7.00371507965759850432377457603, −5.73605759397082683123851971562, −4.94254044807630093027710381598, −3.96566191131626150651477928045, −2.28157624874462099320659148516, −0.871066647531690719624343548300, 2.29906347519646616710003909873, 2.61294405484573639187962449017, 4.43756014944406138822523152988, 5.70845637611590571865827435482, 6.28461101134168519895822415912, 7.16116432541220548462014112801, 8.540341130948359877888310863058, 9.194198574382282196350835653151, 10.12746100893087825046442529963, 10.80111687316540608752738660853

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