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Properties

Label 2-30e2-4.3-c2-0-75
Degree $2$
Conductor $900$
Sign $0.784 + 0.620i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.88 + 0.656i)2-s + (3.13 + 2.48i)4-s − 9.55i·7-s + (4.29 + 6.74i)8-s − 9.92i·11-s − 7.55·13-s + (6.27 − 18.0i)14-s + (3.68 + 15.5i)16-s + 17.1·17-s − 26.1i·19-s + (6.51 − 18.7i)22-s + 1.67i·23-s + (−14.2 − 4.96i)26-s + (23.7 − 29.9i)28-s − 0.350·29-s + ⋯
L(s)  = 1  + (0.944 + 0.328i)2-s + (0.784 + 0.620i)4-s − 1.36i·7-s + (0.537 + 0.843i)8-s − 0.902i·11-s − 0.581·13-s + (0.448 − 1.28i)14-s + (0.230 + 0.973i)16-s + 1.01·17-s − 1.37i·19-s + (0.296 − 0.852i)22-s + 0.0728i·23-s + (−0.549 − 0.190i)26-s + (0.846 − 1.07i)28-s − 0.0120·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.784 + 0.620i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.784 + 0.620i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.330612301\)
\(L(\frac12)\) \(\approx\) \(3.330612301\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.88 - 0.656i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 9.55iT - 49T^{2} \)
11 \( 1 + 9.92iT - 121T^{2} \)
13 \( 1 + 7.55T + 169T^{2} \)
17 \( 1 - 17.1T + 289T^{2} \)
19 \( 1 + 26.1iT - 361T^{2} \)
23 \( 1 - 1.67iT - 529T^{2} \)
29 \( 1 + 0.350T + 841T^{2} \)
31 \( 1 + 46.0iT - 961T^{2} \)
37 \( 1 + 22.6T + 1.36e3T^{2} \)
41 \( 1 - 77.2T + 1.68e3T^{2} \)
43 \( 1 + 41.7iT - 1.84e3T^{2} \)
47 \( 1 - 14.0iT - 2.20e3T^{2} \)
53 \( 1 + 22.6T + 2.80e3T^{2} \)
59 \( 1 - 94.7iT - 3.48e3T^{2} \)
61 \( 1 - 38T + 3.72e3T^{2} \)
67 \( 1 - 29.8iT - 4.48e3T^{2} \)
71 \( 1 + 7.19iT - 5.04e3T^{2} \)
73 \( 1 - 34.3T + 5.32e3T^{2} \)
79 \( 1 + 46.0iT - 6.24e3T^{2} \)
83 \( 1 + 24.1iT - 6.88e3T^{2} \)
89 \( 1 - 100.T + 7.92e3T^{2} \)
97 \( 1 + 131.T + 9.40e3T^{2} \)
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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

−10.01241182150269565994075748699, −8.867812552924653773261550294056, −7.67125724508107715923274943780, −7.34098145207507744554871802076, −6.30693411711675710693847268159, −5.40260031646234628285271007431, −4.40752975383420176452233914924, −3.62052887341594125051596128231, −2.56332271552939179298918475305, −0.798761102314058145546335090377, 1.59198609804141154877198879114, 2.55964484875498927747694772919, 3.57491750341431561966922853526, 4.80781009047372329702471244286, 5.48815527451870452448394860839, 6.27945403391687223708263299311, 7.32429098175362541128046069492, 8.231913762921874316675195302841, 9.481079045750418569655802667551, 9.988788733432053453592214546805

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