Properties

Label 2-2e2-4.3-c20-0-1
Degree $2$
Conductor $4$
Sign $-0.757 + 0.653i$
Analytic cond. $10.1405$
Root an. cond. $3.18442$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (356. + 959. i)2-s + 4.69e4i·3-s + (−7.93e5 + 6.85e5i)4-s − 4.32e6·5-s + (−4.50e7 + 1.67e7i)6-s + 1.10e8i·7-s + (−9.40e8 − 5.17e8i)8-s + 1.28e9·9-s + (−1.54e9 − 4.14e9i)10-s − 2.26e10i·11-s + (−3.21e10 − 3.72e10i)12-s − 2.51e11·13-s + (−1.06e11 + 3.96e10i)14-s − 2.03e11i·15-s + (1.60e11 − 1.08e12i)16-s + 1.40e12·17-s + ⋯
L(s)  = 1  + (0.348 + 0.937i)2-s + 0.795i·3-s + (−0.757 + 0.653i)4-s − 0.442·5-s + (−0.745 + 0.277i)6-s + 0.392i·7-s + (−0.876 − 0.481i)8-s + 0.367·9-s + (−0.154 − 0.414i)10-s − 0.874i·11-s + (−0.519 − 0.602i)12-s − 1.82·13-s + (−0.368 + 0.136i)14-s − 0.352i·15-s + (0.146 − 0.989i)16-s + 0.698·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.757 + 0.653i$
Analytic conductor: \(10.1405\)
Root analytic conductor: \(3.18442\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :10),\ -0.757 + 0.653i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.359421 - 0.966585i\)
\(L(\frac12)\) \(\approx\) \(0.359421 - 0.966585i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-356. - 959. i)T \)
good3 \( 1 - 4.69e4iT - 3.48e9T^{2} \)
5 \( 1 + 4.32e6T + 9.53e13T^{2} \)
7 \( 1 - 1.10e8iT - 7.97e16T^{2} \)
11 \( 1 + 2.26e10iT - 6.72e20T^{2} \)
13 \( 1 + 2.51e11T + 1.90e22T^{2} \)
17 \( 1 - 1.40e12T + 4.06e24T^{2} \)
19 \( 1 - 1.01e13iT - 3.75e25T^{2} \)
23 \( 1 - 4.06e13iT - 1.71e27T^{2} \)
29 \( 1 + 2.78e14T + 1.76e29T^{2} \)
31 \( 1 - 5.16e14iT - 6.71e29T^{2} \)
37 \( 1 - 8.25e14T + 2.31e31T^{2} \)
41 \( 1 + 2.23e14T + 1.80e32T^{2} \)
43 \( 1 + 2.43e16iT - 4.67e32T^{2} \)
47 \( 1 - 7.49e16iT - 2.76e33T^{2} \)
53 \( 1 + 1.39e17T + 3.05e34T^{2} \)
59 \( 1 + 1.92e17iT - 2.61e35T^{2} \)
61 \( 1 - 9.81e17T + 5.08e35T^{2} \)
67 \( 1 + 2.38e18iT - 3.32e36T^{2} \)
71 \( 1 - 6.63e17iT - 1.05e37T^{2} \)
73 \( 1 + 6.58e17T + 1.84e37T^{2} \)
79 \( 1 - 1.44e19iT - 8.96e37T^{2} \)
83 \( 1 - 1.67e19iT - 2.40e38T^{2} \)
89 \( 1 - 3.01e19T + 9.72e38T^{2} \)
97 \( 1 + 3.28e19T + 5.43e39T^{2} \)
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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

−21.28208422553021319702630978613, −18.93351340866361437738829171516, −16.89669867197166998761152370714, −15.69390768788282041034497663919, −14.43433802895363794629578485254, −12.26566065796725782823303552440, −9.676606067465993756321533019617, −7.70190109158113511561664330057, −5.39088766314147944994648701124, −3.69205405194558488466066443716, 0.42913580274664239170127415480, 2.27163407403986887867415056377, 4.56552230274359278857883471555, 7.31039824292693520274591739371, 9.884603930850073100252684118551, 11.91754651885046033395320884109, 13.06403266412629376817712029400, 14.85590585368222215398051363406, 17.57187485901373322468290678435, 19.10198650559788161561745553447

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