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Properties

Label 2-24e2-36.31-c2-0-21
Degree $2$
Conductor $576$
Sign $0.961 + 0.274i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.64 − 1.40i)3-s + (3.01 − 5.22i)5-s + (−10.2 + 5.90i)7-s + (5.04 + 7.45i)9-s + (5.28 − 3.05i)11-s + (−7.44 + 12.9i)13-s + (−15.3 + 9.60i)15-s + 26.6·17-s + 9.45i·19-s + (35.4 − 1.25i)21-s + (17.2 + 9.96i)23-s + (−5.70 − 9.88i)25-s + (−2.86 − 26.8i)27-s + (−22.3 − 38.6i)29-s + (−5.42 − 3.13i)31-s + ⋯
L(s)  = 1  + (−0.883 − 0.469i)3-s + (0.603 − 1.04i)5-s + (−1.46 + 0.844i)7-s + (0.560 + 0.828i)9-s + (0.480 − 0.277i)11-s + (−0.572 + 0.992i)13-s + (−1.02 + 0.640i)15-s + 1.57·17-s + 0.497i·19-s + (1.68 − 0.0597i)21-s + (0.750 + 0.433i)23-s + (−0.228 − 0.395i)25-s + (−0.106 − 0.994i)27-s + (−0.769 − 1.33i)29-s + (−0.174 − 0.101i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.961 + 0.274i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.961 + 0.274i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.203169097\)
\(L(\frac12)\) \(\approx\) \(1.203169097\)
\(L(2)\) 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 \( 1 + (2.64 + 1.40i)T \)
good5 \( 1 + (-3.01 + 5.22i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (10.2 - 5.90i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-5.28 + 3.05i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (7.44 - 12.9i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 26.6T + 289T^{2} \)
19 \( 1 - 9.45iT - 361T^{2} \)
23 \( 1 + (-17.2 - 9.96i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (22.3 + 38.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (5.42 + 3.13i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 6.65T + 1.36e3T^{2} \)
41 \( 1 + (-8.82 + 15.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-20.2 + 11.7i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-36.4 + 21.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 51.6T + 2.80e3T^{2} \)
59 \( 1 + (32.9 + 18.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-45.3 - 78.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-53.4 - 30.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 39.5iT - 5.04e3T^{2} \)
73 \( 1 - 35.0T + 5.32e3T^{2} \)
79 \( 1 + (-77.9 + 45.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-102. + 59.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 14.4T + 7.92e3T^{2} \)
97 \( 1 + (-67.5 - 117. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.31104305928615378625449880355, −9.488793576682256903316702832325, −9.062414587867785629726362118063, −7.67882237451793731371503525832, −6.63747873737733576187244568101, −5.77642777238938025858143882573, −5.32099313071470072686181403153, −3.85260728005231322845206981465, −2.22774204552374398896581662422, −0.870258101063587192590661237635, 0.75305020639984059595381132698, 2.96290411297539713461777921231, 3.68090385960550528805317644526, 5.14318249881817171143807837711, 6.08865982108956115930384834053, 6.83680486671959035625296456235, 7.43674725185701745482373916482, 9.335700229416382207305826286435, 9.876055370360016888083183218419, 10.49304914258576985515571663669

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