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Properties

Label 2-2e2-4.3-c38-0-6
Degree $2$
Conductor $4$
Sign $0.907 + 0.419i$
Analytic cond. $36.5853$
Root an. cond. $6.04858$
Motivic weight $38$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.12e5 − 5.12e5i)2-s + 1.96e9i·3-s + (−2.49e11 − 1.15e11i)4-s − 3.09e13·5-s + (1.00e15 + 2.21e14i)6-s − 9.14e15i·7-s + (−8.71e16 + 1.14e17i)8-s − 2.50e18·9-s + (−3.48e18 + 1.58e19i)10-s − 4.11e19i·11-s + (2.26e20 − 4.89e20i)12-s − 1.68e21·13-s + (−4.68e21 − 1.02e21i)14-s − 6.07e22i·15-s + (4.89e22 + 5.75e22i)16-s + 6.20e22·17-s + ⋯
L(s)  = 1  + (0.214 − 0.976i)2-s + 1.68i·3-s + (−0.907 − 0.419i)4-s − 1.62·5-s + (1.64 + 0.362i)6-s − 0.802i·7-s + (−0.604 + 0.796i)8-s − 1.85·9-s + (−0.348 + 1.58i)10-s − 0.673i·11-s + (0.708 − 1.53i)12-s − 1.15·13-s + (−0.783 − 0.172i)14-s − 2.74i·15-s + (0.648 + 0.761i)16-s + 0.259·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.907 + 0.419i$
Analytic conductor: \(36.5853\)
Root analytic conductor: \(6.04858\)
Motivic weight: \(38\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :19),\ 0.907 + 0.419i)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(0.8254847808\)
\(L(\frac12)\) \(\approx\) \(0.8254847808\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.12e5 + 5.12e5i)T \)
good3 \( 1 - 1.96e9iT - 1.35e18T^{2} \)
5 \( 1 + 3.09e13T + 3.63e26T^{2} \)
7 \( 1 + 9.14e15iT - 1.29e32T^{2} \)
11 \( 1 + 4.11e19iT - 3.74e39T^{2} \)
13 \( 1 + 1.68e21T + 2.13e42T^{2} \)
17 \( 1 - 6.20e22T + 5.71e46T^{2} \)
19 \( 1 - 1.64e24iT - 3.91e48T^{2} \)
23 \( 1 - 7.68e25iT - 5.56e51T^{2} \)
29 \( 1 - 9.19e27T + 3.72e55T^{2} \)
31 \( 1 + 1.50e28iT - 4.69e56T^{2} \)
37 \( 1 + 3.34e29T + 3.90e59T^{2} \)
41 \( 1 + 4.03e29T + 1.93e61T^{2} \)
43 \( 1 - 9.22e30iT - 1.17e62T^{2} \)
47 \( 1 + 8.34e31iT - 3.46e63T^{2} \)
53 \( 1 + 8.95e32T + 3.33e65T^{2} \)
59 \( 1 + 2.84e33iT - 1.96e67T^{2} \)
61 \( 1 - 5.02e33T + 6.95e67T^{2} \)
67 \( 1 + 3.32e34iT - 2.45e69T^{2} \)
71 \( 1 - 1.07e33iT - 2.22e70T^{2} \)
73 \( 1 - 4.49e35T + 6.40e70T^{2} \)
79 \( 1 + 2.34e35iT - 1.28e72T^{2} \)
83 \( 1 - 3.08e36iT - 8.41e72T^{2} \)
89 \( 1 + 4.93e36T + 1.19e74T^{2} \)
97 \( 1 - 3.62e37T + 3.14e75T^{2} \)
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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

−15.51434441750371754240118723281, −14.29612556513852962452957024927, −11.95364255585572154876776023638, −10.89569063260163096214485872103, −9.787982447741362681025447328676, −8.132036933423025032586157120342, −4.97612616504818042478518286326, −3.96616256373016499995854797447, −3.24721192773905434013514303850, −0.45802665497116310681613719135, 0.57910890815766653152658628481, 2.74255685999802405772369108960, 4.79440512117220800677472371498, 6.69721186361533398468756644039, 7.55328541070358570954674976768, 8.568316908779569397621223574447, 12.08044381424259670324378044111, 12.51907957596018543892089256747, 14.42116413194391584877181738030, 15.66587065856079882104760851888

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