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Properties

Label 2-22e2-11.5-c1-0-0
Degree $2$
Conductor $484$
Sign $0.437 - 0.899i$
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 − 0.587i)3-s + (−0.927 + 2.85i)5-s + (1.61 − 1.17i)7-s + (−0.618 − 1.90i)9-s + (1.23 + 3.80i)13-s + (2.42 − 1.76i)15-s + (−1.85 + 5.70i)17-s + (6.47 + 4.70i)19-s − 2·21-s − 3·23-s + (−3.23 − 2.35i)25-s + (−1.54 + 4.75i)27-s + (1.54 + 4.75i)31-s + (1.85 + 5.70i)35-s + (0.809 − 0.587i)37-s + ⋯
L(s)  = 1  + (−0.467 − 0.339i)3-s + (−0.414 + 1.27i)5-s + (0.611 − 0.444i)7-s + (−0.206 − 0.634i)9-s + (0.342 + 1.05i)13-s + (0.626 − 0.455i)15-s + (−0.449 + 1.38i)17-s + (1.48 + 1.07i)19-s − 0.436·21-s − 0.625·23-s + (−0.647 − 0.470i)25-s + (−0.297 + 0.915i)27-s + (0.277 + 0.854i)31-s + (0.313 + 0.964i)35-s + (0.133 − 0.0966i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(3.86475\)
Root analytic conductor: \(1.96589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :1/2),\ 0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917858 + 0.573898i\)
\(L(\frac12)\) \(\approx\) \(0.917858 + 0.573898i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.927 - 2.85i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.61 + 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.23 - 3.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.85 - 5.70i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.47 - 4.70i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.54 - 4.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.85 + 5.70i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.42 - 1.76i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.23 + 3.80i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 + (-4.63 + 14.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.23 - 2.35i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.618 + 1.90i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.85 - 5.70i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (2.16 + 6.65i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.15679624540527634512232008309, −10.54294040774090417440630834271, −9.461710514557496028302263049568, −8.236239476541034713312465163456, −7.35282880578821108307230286617, −6.56518263730742926592386490519, −5.80503758436455896158296472628, −4.18844930802716877080359787905, −3.32869625136823597935180608635, −1.58062411332497186561083618198, 0.75657274691484477852318630432, 2.66317511789158502783548007910, 4.38964446451887250315531541308, 5.12014718249656613689065079337, 5.69060491927036264727887658014, 7.42396389834517857343184190716, 8.155017022335705648214371107394, 9.002696220187920954717621838272, 9.851736772060427031113255680687, 11.11871204488279129646591513574

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