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Properties

Label 2-2e2-4.3-c30-0-10
Degree $2$
Conductor $4$
Sign $0.790 - 0.612i$
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  + (1.06e4 + 3.10e4i)2-s + 2.79e6i·3-s + (−8.48e8 + 6.57e8i)4-s + 4.95e10·5-s + (−8.66e10 + 2.96e10i)6-s − 4.58e12i·7-s + (−2.94e13 − 1.93e13i)8-s + 1.98e14·9-s + (5.25e14 + 1.53e15i)10-s − 7.94e15i·11-s + (−1.83e15 − 2.37e15i)12-s + 4.95e16·13-s + (1.42e17 − 4.86e16i)14-s + 1.38e17i·15-s + (2.87e17 − 1.11e18i)16-s − 1.14e18·17-s + ⋯
L(s)  = 1  + (0.323 + 0.946i)2-s + 0.194i·3-s + (−0.790 + 0.612i)4-s + 1.62·5-s + (−0.184 + 0.0631i)6-s − 0.965i·7-s + (−0.835 − 0.549i)8-s + 0.962·9-s + (0.525 + 1.53i)10-s − 1.90i·11-s + (−0.119 − 0.154i)12-s + 0.968·13-s + (0.913 − 0.312i)14-s + 0.316i·15-s + (0.249 − 0.968i)16-s − 0.398·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.790 - 0.612i$
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: \((2,\ 4,\ (\ :15),\ 0.790 - 0.612i)\)

Particular Values

\(L(\frac{31}{2})\) \(\approx\) \(2.81384 + 0.963047i\)
\(L(\frac12)\) \(\approx\) \(2.81384 + 0.963047i\)
\(L(16)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06e4 - 3.10e4i)T \)
good3 \( 1 - 2.79e6iT - 2.05e14T^{2} \)
5 \( 1 - 4.95e10T + 9.31e20T^{2} \)
7 \( 1 + 4.58e12iT - 2.25e25T^{2} \)
11 \( 1 + 7.94e15iT - 1.74e31T^{2} \)
13 \( 1 - 4.95e16T + 2.61e33T^{2} \)
17 \( 1 + 1.14e18T + 8.19e36T^{2} \)
19 \( 1 + 3.03e17iT - 2.30e38T^{2} \)
23 \( 1 - 2.22e20iT - 7.10e40T^{2} \)
29 \( 1 - 2.57e21T + 7.44e43T^{2} \)
31 \( 1 - 2.49e22iT - 5.50e44T^{2} \)
37 \( 1 + 2.12e23T + 1.11e47T^{2} \)
41 \( 1 + 1.08e24T + 2.41e48T^{2} \)
43 \( 1 + 4.37e24iT - 1.00e49T^{2} \)
47 \( 1 + 5.21e24iT - 1.45e50T^{2} \)
53 \( 1 - 6.30e25T + 5.34e51T^{2} \)
59 \( 1 + 4.44e25iT - 1.33e53T^{2} \)
61 \( 1 + 5.73e26T + 3.62e53T^{2} \)
67 \( 1 - 1.06e27iT - 6.05e54T^{2} \)
71 \( 1 - 8.31e27iT - 3.44e55T^{2} \)
73 \( 1 + 7.05e26T + 7.93e55T^{2} \)
79 \( 1 - 4.98e27iT - 8.48e56T^{2} \)
83 \( 1 - 5.59e28iT - 3.73e57T^{2} \)
89 \( 1 - 1.95e29T + 3.03e58T^{2} \)
97 \( 1 - 4.93e29T + 4.01e59T^{2} \)
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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

−17.15220877332140931938609445646, −15.99698967091444394195035212008, −13.80594445342879125168727514076, −13.43955428088627762924498272238, −10.41650945237272409910979310821, −8.810929962026112407387034840784, −6.72312301927462025904558681791, −5.49156845192033489888250596789, −3.61813598725403433934774828183, −1.07814525830191649607035030700, 1.56156658322239310380327328511, 2.30023310661525568401436291027, 4.69478485062036062446262200665, 6.24284518277121319605133001185, 9.237636705108428750859087380636, 10.24501001941973859241012854421, 12.40942455588164268707736944928, 13.39079513269212941738300809494, 15.06028277062564890967777021869, 17.78984990798931132499667303622

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