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Properties

Label 2-2e2-4.3-c10-0-3
Degree $2$
Conductor $4$
Sign $0.259 + 0.965i$
Analytic cond. $2.54142$
Root an. cond. $1.59418$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (19.4 − 25.3i)2-s − 120. i·3-s + (−265. − 988. i)4-s + 2.48e3·5-s + (−3.05e3 − 2.34e3i)6-s + 2.91e4i·7-s + (−3.02e4 − 1.25e4i)8-s + 4.45e4·9-s + (4.84e4 − 6.31e4i)10-s − 8.84e4i·11-s + (−1.18e5 + 3.19e4i)12-s − 3.00e5·13-s + (7.39e5 + 5.67e5i)14-s − 2.99e5i·15-s + (−9.07e5 + 5.25e5i)16-s − 1.52e5·17-s + ⋯
L(s)  = 1  + (0.608 − 0.793i)2-s − 0.494i·3-s + (−0.259 − 0.965i)4-s + 0.795·5-s + (−0.392 − 0.301i)6-s + 1.73i·7-s + (−0.924 − 0.381i)8-s + 0.755·9-s + (0.484 − 0.631i)10-s − 0.549i·11-s + (−0.477 + 0.128i)12-s − 0.809·13-s + (1.37 + 1.05i)14-s − 0.393i·15-s + (−0.865 + 0.501i)16-s − 0.107·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.259 + 0.965i$
Analytic conductor: \(2.54142\)
Root analytic conductor: \(1.59418\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :5),\ 0.259 + 0.965i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.44565 - 1.10855i\)
\(L(\frac12)\) \(\approx\) \(1.44565 - 1.10855i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-19.4 + 25.3i)T \)
good3 \( 1 + 120. iT - 5.90e4T^{2} \)
5 \( 1 - 2.48e3T + 9.76e6T^{2} \)
7 \( 1 - 2.91e4iT - 2.82e8T^{2} \)
11 \( 1 + 8.84e4iT - 2.59e10T^{2} \)
13 \( 1 + 3.00e5T + 1.37e11T^{2} \)
17 \( 1 + 1.52e5T + 2.01e12T^{2} \)
19 \( 1 - 4.76e4iT - 6.13e12T^{2} \)
23 \( 1 - 7.60e6iT - 4.14e13T^{2} \)
29 \( 1 + 6.49e6T + 4.20e14T^{2} \)
31 \( 1 + 3.06e7iT - 8.19e14T^{2} \)
37 \( 1 - 5.72e7T + 4.80e15T^{2} \)
41 \( 1 + 2.49e7T + 1.34e16T^{2} \)
43 \( 1 + 1.22e8iT - 2.16e16T^{2} \)
47 \( 1 + 1.03e7iT - 5.25e16T^{2} \)
53 \( 1 - 5.65e8T + 1.74e17T^{2} \)
59 \( 1 - 3.70e8iT - 5.11e17T^{2} \)
61 \( 1 - 3.89e8T + 7.13e17T^{2} \)
67 \( 1 + 1.36e9iT - 1.82e18T^{2} \)
71 \( 1 + 1.93e9iT - 3.25e18T^{2} \)
73 \( 1 + 3.26e9T + 4.29e18T^{2} \)
79 \( 1 + 2.38e8iT - 9.46e18T^{2} \)
83 \( 1 - 6.66e9iT - 1.55e19T^{2} \)
89 \( 1 - 2.76e9T + 3.11e19T^{2} \)
97 \( 1 - 1.47e9T + 7.37e19T^{2} \)
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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

−22.03249571383238934986586655325, −21.36507386070773267698383403046, −19.19146084245894350801067916891, −18.12988880105713957401230650970, −15.21488031710711449531718585184, −13.33794713020308486638585085897, −11.94507792642790259188233467505, −9.529020991482436045869075689001, −5.72392889532439134048585383642, −2.14674351421075907605324664010, 4.43798841756105197179041122545, 7.10839456140413286653479066916, 10.05123483445908925802978314686, 13.08228593701896098298789512989, 14.51262165287269867087373644324, 16.42444895783949711496739066423, 17.59707180947016289892955724082, 20.47028879302976326012815410521, 21.79510207899135475795331149790, 23.22664503655558199168076198860

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