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Properties

Label 2-2e2-4.3-c22-0-4
Degree $2$
Conductor $4$
Sign $0.988 - 0.153i$
Analytic cond. $12.2682$
Root an. cond. $3.50261$
Motivic weight $22$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (157. + 2.04e3i)2-s − 7.57e4i·3-s + (−4.14e6 + 6.42e5i)4-s + 1.73e7·5-s + (1.54e8 − 1.19e7i)6-s − 2.02e9i·7-s + (−1.96e9 − 8.36e9i)8-s + 2.56e10·9-s + (2.72e9 + 3.53e10i)10-s + 1.41e11i·11-s + (4.86e10 + 3.13e11i)12-s + 2.02e12·13-s + (4.14e12 − 3.19e11i)14-s − 1.31e12i·15-s + (1.67e13 − 5.32e12i)16-s + 3.17e13·17-s + ⋯
L(s)  = 1  + (0.0767 + 0.997i)2-s − 0.427i·3-s + (−0.988 + 0.153i)4-s + 0.354·5-s + (0.426 − 0.0328i)6-s − 1.02i·7-s + (−0.228 − 0.973i)8-s + 0.817·9-s + (0.0272 + 0.353i)10-s + 0.494i·11-s + (0.0654 + 0.422i)12-s + 1.12·13-s + (1.02 − 0.0787i)14-s − 0.151i·15-s + (0.953 − 0.302i)16-s + 0.926·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.988 - 0.153i$
Analytic conductor: \(12.2682\)
Root analytic conductor: \(3.50261\)
Motivic weight: \(22\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :11),\ 0.988 - 0.153i)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(1.87953 + 0.144701i\)
\(L(\frac12)\) \(\approx\) \(1.87953 + 0.144701i\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-157. - 2.04e3i)T \)
good3 \( 1 + 7.57e4iT - 3.13e10T^{2} \)
5 \( 1 - 1.73e7T + 2.38e15T^{2} \)
7 \( 1 + 2.02e9iT - 3.90e18T^{2} \)
11 \( 1 - 1.41e11iT - 8.14e22T^{2} \)
13 \( 1 - 2.02e12T + 3.21e24T^{2} \)
17 \( 1 - 3.17e13T + 1.17e27T^{2} \)
19 \( 1 - 6.95e12iT - 1.35e28T^{2} \)
23 \( 1 + 1.28e15iT - 9.07e29T^{2} \)
29 \( 1 - 1.62e16T + 1.48e32T^{2} \)
31 \( 1 + 4.83e16iT - 6.45e32T^{2} \)
37 \( 1 + 2.07e17T + 3.16e34T^{2} \)
41 \( 1 - 3.01e17T + 3.02e35T^{2} \)
43 \( 1 - 1.19e18iT - 8.63e35T^{2} \)
47 \( 1 - 3.29e18iT - 6.11e36T^{2} \)
53 \( 1 + 5.81e18T + 8.59e37T^{2} \)
59 \( 1 + 3.91e19iT - 9.09e38T^{2} \)
61 \( 1 - 2.65e19T + 1.89e39T^{2} \)
67 \( 1 - 7.00e19iT - 1.49e40T^{2} \)
71 \( 1 - 8.53e19iT - 5.34e40T^{2} \)
73 \( 1 + 2.36e20T + 9.84e40T^{2} \)
79 \( 1 - 1.08e21iT - 5.59e41T^{2} \)
83 \( 1 + 1.04e21iT - 1.65e42T^{2} \)
89 \( 1 + 2.13e21T + 7.70e42T^{2} \)
97 \( 1 - 8.47e21T + 5.11e43T^{2} \)
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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

−18.77285247848163624549665435792, −17.45987897254300137323632034011, −15.99365529107064481959347249897, −14.11269738471693929133070473250, −12.89248289515643312200216727506, −10.00361650111839934715841999975, −7.82597942189678470540426252314, −6.38758019192599583431488740162, −4.20706632793378378306353569790, −0.986170029005959834050395739816, 1.48196965519575721850945787507, 3.44764334670283408639561511489, 5.46265427853766153489221041885, 8.824061419984298007590192802387, 10.31699815797111589076797343640, 12.06505975394585706955495499223, 13.73196975239418058227355316754, 15.70851695682900426777197355842, 17.97897373107672116246336341199, 19.18126020488685072302416266304

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