Properties

Label 2-2e2-4.3-c6-0-1
Degree $2$
Conductor $4$
Sign $0.875 + 0.484i$
Analytic cond. $0.920216$
Root an. cond. $0.959279$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 7.74i)2-s + 30.9i·3-s + (−56.0 − 30.9i)4-s + 10·5-s + (240. + 61.9i)6-s − 309. i·7-s + (−352. + 371. i)8-s − 231.·9-s + (20 − 77.4i)10-s + 960. i·11-s + (960. − 1.73e3i)12-s + 1.46e3·13-s + (−2.40e3 − 619. i)14-s + 309. i·15-s + (2.17e3 + 3.47e3i)16-s − 4.76e3·17-s + ⋯
L(s)  = 1  + (0.250 − 0.968i)2-s + 1.14i·3-s + (−0.875 − 0.484i)4-s + 0.0800·5-s + (1.11 + 0.286i)6-s − 0.903i·7-s + (−0.687 + 0.726i)8-s − 0.316·9-s + (0.0200 − 0.0774i)10-s + 0.721i·11-s + (0.555 − 1.00i)12-s + 0.667·13-s + (−0.874 − 0.225i)14-s + 0.0918i·15-s + (0.531 + 0.847i)16-s − 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.484i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.875 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.875 + 0.484i$
Analytic conductor: \(0.920216\)
Root analytic conductor: \(0.959279\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :3),\ 0.875 + 0.484i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.00282 - 0.258927i\)
\(L(\frac12)\) \(\approx\) \(1.00282 - 0.258927i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 7.74i)T \)
good3 \( 1 - 30.9iT - 729T^{2} \)
5 \( 1 - 10T + 1.56e4T^{2} \)
7 \( 1 + 309. iT - 1.17e5T^{2} \)
11 \( 1 - 960. iT - 1.77e6T^{2} \)
13 \( 1 - 1.46e3T + 4.82e6T^{2} \)
17 \( 1 + 4.76e3T + 2.41e7T^{2} \)
19 \( 1 + 7.52e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.04e4iT - 1.48e8T^{2} \)
29 \( 1 - 2.54e4T + 5.94e8T^{2} \)
31 \( 1 + 4.18e4iT - 8.87e8T^{2} \)
37 \( 1 - 1.99e3T + 2.56e9T^{2} \)
41 \( 1 - 2.93e4T + 4.75e9T^{2} \)
43 \( 1 - 2.15e4iT - 6.32e9T^{2} \)
47 \( 1 - 7.56e3iT - 1.07e10T^{2} \)
53 \( 1 + 1.92e5T + 2.21e10T^{2} \)
59 \( 1 - 7.84e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.09e4T + 5.15e10T^{2} \)
67 \( 1 - 3.94e5iT - 9.04e10T^{2} \)
71 \( 1 + 5.32e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.88e5T + 1.51e11T^{2} \)
79 \( 1 - 3.10e5iT - 2.43e11T^{2} \)
83 \( 1 - 2.04e5iT - 3.26e11T^{2} \)
89 \( 1 - 3.10e5T + 4.96e11T^{2} \)
97 \( 1 + 1.45e6T + 8.32e11T^{2} \)
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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

−23.57175299273089970011604651517, −22.19059552843461559609099182859, −20.95756433816106989269764042540, −19.83058320096091245143230269053, −17.59929793768923017837605476704, −15.43249017370527487601311095726, −13.47879660771318523930785751614, −11.03139954962341675621445453554, −9.604181779505845522978754643876, −4.29313366391969956478778604249, 6.27799477609798837982289869787, 8.405706884823255524905660661091, 12.41897331860520707654073360983, 13.89484124930942548130012896258, 15.87856099392868848643883166863, 17.88102778607635228745883699780, 18.90580774121245845310976603894, 21.61858622259975424842452197345, 23.26626467501858284215669449273, 24.53660273464275639743622462145

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