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Properties

Label 2-22e2-11.4-c1-0-2
Degree $2$
Conductor $484$
Sign $-0.530 - 0.847i$
Analytic cond. $3.86475$
Root an. cond. $1.96589$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 2.48i)3-s + (1.30 − 0.951i)5-s + (1.19 + 3.66i)7-s + (−3.11 − 2.26i)9-s + (1.92 + 1.40i)13-s + (1.30 + 4.02i)15-s + (1.92 − 1.40i)17-s + (−1.19 + 3.66i)19-s − 10.0·21-s − 2.47·23-s + (−0.736 + 2.26i)25-s + (1.80 − 1.31i)27-s + (−2.66 − 8.19i)29-s + (−0.690 − 0.502i)31-s + (5.04 + 3.66i)35-s + ⋯
L(s)  = 1  + (−0.467 + 1.43i)3-s + (0.585 − 0.425i)5-s + (0.450 + 1.38i)7-s + (−1.03 − 0.755i)9-s + (0.534 + 0.388i)13-s + (0.337 + 1.04i)15-s + (0.467 − 0.339i)17-s + (−0.273 + 0.840i)19-s − 2.20·21-s − 0.515·23-s + (−0.147 + 0.453i)25-s + (0.348 − 0.252i)27-s + (−0.494 − 1.52i)29-s + (−0.124 − 0.0901i)31-s + (0.852 + 0.619i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $-0.530 - 0.847i$
Analytic conductor: \(3.86475\)
Root analytic conductor: \(1.96589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :1/2),\ -0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.638521 + 1.15230i\)
\(L(\frac12)\) \(\approx\) \(0.638521 + 1.15230i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.809 - 2.48i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.30 + 0.951i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.19 - 3.66i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.92 - 1.40i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.92 + 1.40i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.19 - 3.66i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + (2.66 + 8.19i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.690 + 0.502i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.572 + 1.76i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.66 - 8.19i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-0.427 + 1.31i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.30 - 2.40i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.336 - 1.03i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.92 + 1.40i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + (-5.16 + 3.75i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.281 - 0.865i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.78 + 4.20i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.5 + 7.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.472T + 89T^{2} \)
97 \( 1 + (-11.7 - 8.55i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.36889184451215961412180185360, −10.21260498023722732551512060975, −9.553300732954448761024702264402, −8.915274851079268624966294248502, −7.971831545909616965577349615314, −6.09398667100770180032193753771, −5.59140007141966739848062126577, −4.74396113408630387427084492874, −3.63780121958315386093209429462, −2.03054815940115123780382073172, 0.890890576571586696560683311209, 2.05828146868527219068607943632, 3.69800636310515816265694489778, 5.21637355879491133525905772197, 6.28391953474641100465568500152, 6.99020009699597258661366239713, 7.65778654935867587374017488855, 8.621644805833393307978319747476, 10.11070645038356005015816054633, 10.75995237817235674501413115238

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