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Properties

Label 2-22e2-11.4-c1-0-7
Degree $2$
Conductor $484$
Sign $-0.975 + 0.220i$
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  + (0.618 − 1.90i)3-s + (−2.42 + 1.76i)5-s + (−1.07 − 3.29i)7-s + (−0.809 − 0.587i)9-s + (−4.20 − 3.05i)13-s + (1.85 + 5.70i)15-s + (−4.20 + 3.05i)17-s + (1.07 − 3.29i)19-s − 6.92·21-s − 6·23-s + (1.23 − 3.80i)25-s + (3.23 − 2.35i)27-s + (−1.60 − 4.94i)29-s + (1.61 + 1.17i)31-s + (8.40 + 6.10i)35-s + ⋯
L(s)  = 1  + (0.356 − 1.09i)3-s + (−1.08 + 0.788i)5-s + (−0.404 − 1.24i)7-s + (−0.269 − 0.195i)9-s + (−1.16 − 0.847i)13-s + (0.478 + 1.47i)15-s + (−1.01 + 0.740i)17-s + (0.245 − 0.755i)19-s − 1.51·21-s − 1.25·23-s + (0.247 − 0.760i)25-s + (0.622 − 0.452i)27-s + (−0.298 − 0.917i)29-s + (0.290 + 0.211i)31-s + (1.42 + 1.03i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $-0.975 + 0.220i$
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: \((2,\ 484,\ (\ :1/2),\ -0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0728288 - 0.651908i\)
\(L(\frac12)\) \(\approx\) \(0.0728288 - 0.651908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 + 1.90i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.42 - 1.76i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.07 + 3.29i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.20 + 3.05i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.20 - 3.05i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.07 + 3.29i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (1.60 + 4.94i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.61 - 1.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.60 - 4.94i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-1.85 + 5.70i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.42 - 1.76i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-11.2 + 8.14i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.14 + 6.58i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.80 + 2.03i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.40 + 6.10i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (4.04 + 2.93i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.57452885704478619954083217910, −9.902811524831733221658861561031, −8.321831380003533144693634970550, −7.61927707829042791593551834654, −7.13977268665294810624635897971, −6.36523027303624625484282957568, −4.53866992888033644827080646767, −3.52235981530607976794156224896, −2.32333578395404876215973629464, −0.36148131048104681031669543136, 2.43121871883488838962939537375, 3.80524034706204569090963199660, 4.55101552156429345134867232519, 5.44528180839110309185690222641, 6.88978409245489273468400204120, 8.050559493354519739151629485784, 8.999547538451491234567072430202, 9.344036891858636104214463272551, 10.30804182061234352392479610720, 11.64563345249898489616272164366

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