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Properties

Label 2-2e2-4.3-c30-0-9
Degree $2$
Conductor $4$
Sign $0.0784 + 0.996i$
Analytic cond. $22.8057$
Root an. cond. $4.77553$
Motivic weight $30$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.22e4 − 2.40e4i)2-s + 2.23e7i·3-s + (−8.42e7 − 1.07e9i)4-s − 2.29e10·5-s + (5.36e11 + 4.96e11i)6-s − 1.19e12i·7-s + (−2.76e13 − 2.17e13i)8-s − 2.91e14·9-s + (−5.10e14 + 5.51e14i)10-s − 4.93e15i·11-s + (2.38e16 − 1.87e15i)12-s + 8.97e16·13-s + (−2.88e16 − 2.66e16i)14-s − 5.11e17i·15-s + (−1.13e18 + 1.80e17i)16-s + 3.32e18·17-s + ⋯
L(s)  = 1  + (0.678 − 0.734i)2-s + 1.55i·3-s + (−0.0784 − 0.996i)4-s − 0.751·5-s + (1.14 + 1.05i)6-s − 0.252i·7-s + (−0.785 − 0.619i)8-s − 1.41·9-s + (−0.510 + 0.551i)10-s − 1.18i·11-s + (1.54 − 0.121i)12-s + 1.75·13-s + (−0.185 − 0.171i)14-s − 1.16i·15-s + (−0.987 + 0.156i)16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0784 + 0.996i)\, \overline{\Lambda}(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & (0.0784 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.0784 + 0.996i$
Analytic conductor: \(22.8057\)
Root analytic conductor: \(4.77553\)
Motivic weight: \(30\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :15),\ 0.0784 + 0.996i)\)

Particular Values

\(L(\frac{31}{2})\) \(\approx\) \(1.45695 - 1.34683i\)
\(L(\frac12)\) \(\approx\) \(1.45695 - 1.34683i\)
\(L(16)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.22e4 + 2.40e4i)T \)
good3 \( 1 - 2.23e7iT - 2.05e14T^{2} \)
5 \( 1 + 2.29e10T + 9.31e20T^{2} \)
7 \( 1 + 1.19e12iT - 2.25e25T^{2} \)
11 \( 1 + 4.93e15iT - 1.74e31T^{2} \)
13 \( 1 - 8.97e16T + 2.61e33T^{2} \)
17 \( 1 - 3.32e18T + 8.19e36T^{2} \)
19 \( 1 + 1.94e19iT - 2.30e38T^{2} \)
23 \( 1 + 1.69e20iT - 7.10e40T^{2} \)
29 \( 1 + 4.31e21T + 7.44e43T^{2} \)
31 \( 1 + 2.81e22iT - 5.50e44T^{2} \)
37 \( 1 + 5.30e23T + 1.11e47T^{2} \)
41 \( 1 - 1.22e24T + 2.41e48T^{2} \)
43 \( 1 + 2.21e24iT - 1.00e49T^{2} \)
47 \( 1 + 5.35e24iT - 1.45e50T^{2} \)
53 \( 1 + 2.57e25T + 5.34e51T^{2} \)
59 \( 1 - 4.55e26iT - 1.33e53T^{2} \)
61 \( 1 - 4.85e26T + 3.62e53T^{2} \)
67 \( 1 + 1.53e27iT - 6.05e54T^{2} \)
71 \( 1 + 4.87e27iT - 3.44e55T^{2} \)
73 \( 1 + 1.12e28T + 7.93e55T^{2} \)
79 \( 1 - 7.24e26iT - 8.48e56T^{2} \)
83 \( 1 - 2.29e28iT - 3.73e57T^{2} \)
89 \( 1 - 1.20e29T + 3.03e58T^{2} \)
97 \( 1 + 5.45e29T + 4.01e59T^{2} \)
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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

−16.25935809345973850775820959739, −15.30344131120855716454218925488, −13.71383515371737685356695191154, −11.44211976857702433951927541870, −10.57184504740800409845184637444, −8.867640473659257038924482437688, −5.67259061274488853634885533722, −4.06527176950662281464143034674, −3.32507947119376303688436073393, −0.58440785191142448839047639949, 1.54167435546098276669630813328, 3.59406682790909835561701720957, 5.84066843000064460966253563128, 7.25345179009122211435654064950, 8.237920988386611247788116814030, 11.86857122032310710933305197718, 12.77145172991186739727364214824, 14.21224811035143960230049220039, 15.87334184809186366887814869414, 17.71052266650213805216110339616

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