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Properties

Label 2-2e2-4.3-c18-0-7
Degree $2$
Conductor $4$
Sign $-0.736 + 0.676i$
Analytic cond. $8.21544$
Root an. cond. $2.86625$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (477. − 185. i)2-s − 2.70e4i·3-s + (1.93e5 − 1.77e5i)4-s − 2.11e6·5-s + (−5.03e6 − 1.29e7i)6-s + 3.35e7i·7-s + (5.92e7 − 1.20e8i)8-s − 3.46e8·9-s + (−1.01e9 + 3.93e8i)10-s − 3.87e9i·11-s + (−4.80e9 − 5.23e9i)12-s + 7.06e8·13-s + (6.23e9 + 1.60e10i)14-s + 5.74e10i·15-s + (5.90e9 − 6.84e10i)16-s + 2.14e11·17-s + ⋯
L(s)  = 1  + (0.931 − 0.362i)2-s − 1.37i·3-s + (0.736 − 0.676i)4-s − 1.08·5-s + (−0.499 − 1.28i)6-s + 0.831i·7-s + (0.441 − 0.897i)8-s − 0.894·9-s + (−1.01 + 0.393i)10-s − 1.64i·11-s + (−0.930 − 1.01i)12-s + 0.0665·13-s + (0.301 + 0.774i)14-s + 1.49i·15-s + (0.0858 − 0.996i)16-s + 1.81·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.736 + 0.676i$
Analytic conductor: \(8.21544\)
Root analytic conductor: \(2.86625\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :9),\ -0.736 + 0.676i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.859268 - 2.20749i\)
\(L(\frac12)\) \(\approx\) \(0.859268 - 2.20749i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-477. + 185. i)T \)
good3 \( 1 + 2.70e4iT - 3.87e8T^{2} \)
5 \( 1 + 2.11e6T + 3.81e12T^{2} \)
7 \( 1 - 3.35e7iT - 1.62e15T^{2} \)
11 \( 1 + 3.87e9iT - 5.55e18T^{2} \)
13 \( 1 - 7.06e8T + 1.12e20T^{2} \)
17 \( 1 - 2.14e11T + 1.40e22T^{2} \)
19 \( 1 + 5.73e9iT - 1.04e23T^{2} \)
23 \( 1 - 1.10e12iT - 3.24e24T^{2} \)
29 \( 1 - 1.21e13T + 2.10e26T^{2} \)
31 \( 1 - 5.37e12iT - 6.99e26T^{2} \)
37 \( 1 + 2.74e13T + 1.68e28T^{2} \)
41 \( 1 - 2.53e14T + 1.07e29T^{2} \)
43 \( 1 - 3.34e14iT - 2.52e29T^{2} \)
47 \( 1 - 1.35e15iT - 1.25e30T^{2} \)
53 \( 1 + 3.47e15T + 1.08e31T^{2} \)
59 \( 1 + 3.11e15iT - 7.50e31T^{2} \)
61 \( 1 + 6.55e15T + 1.36e32T^{2} \)
67 \( 1 + 8.23e15iT - 7.40e32T^{2} \)
71 \( 1 - 7.54e16iT - 2.10e33T^{2} \)
73 \( 1 - 9.58e15T + 3.46e33T^{2} \)
79 \( 1 + 1.19e17iT - 1.43e34T^{2} \)
83 \( 1 - 1.86e17iT - 3.49e34T^{2} \)
89 \( 1 - 1.85e17T + 1.22e35T^{2} \)
97 \( 1 - 7.14e17T + 5.77e35T^{2} \)
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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

−19.35914563053143045926712953981, −18.82194797828805241595354965278, −15.95707541059625132977699214521, −14.09911598901124001459567102370, −12.48731644909282166916345360931, −11.51600816648073989413452280196, −7.85408510370928911101879306299, −5.92743960769429963686139471082, −3.14026800099141063940624539318, −0.994155710131953499849212578784, 3.67606704564291121445981407910, 4.70403803305665355441455857565, 7.52432762455662688913262090481, 10.32650744788821474012740838065, 12.16103381134670891384900938670, 14.57985366953404570191153937811, 15.64687175239783165034213087019, 16.83009800257283918117799087239, 20.07112671489246898832667157659, 21.02893579015721028308937053921

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