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Properties

Label 2-6e4-9.5-c2-0-42
Degree $2$
Conductor $1296$
Sign $-0.0871 + 0.996i$
Analytic cond. $35.3134$
Root an. cond. $5.94251$
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  + (5.01 + 2.89i)5-s + (−4.19 − 7.26i)7-s + (12.7 − 7.34i)11-s + (10.5 − 18.3i)13-s − 7.76i·17-s − 24.3·19-s + (12.7 + 7.34i)23-s + (4.29 + 7.43i)25-s + (−30.7 + 17.7i)29-s + (4 − 6.92i)31-s − 48.6i·35-s − 60.5·37-s + (−29.1 − 16.8i)41-s + (4.58 + 7.94i)43-s + (−14.6 + 8.48i)47-s + ⋯
L(s)  = 1  + (1.00 + 0.579i)5-s + (−0.599 − 1.03i)7-s + (1.15 − 0.668i)11-s + (0.815 − 1.41i)13-s − 0.456i·17-s − 1.28·19-s + (0.553 + 0.319i)23-s + (0.171 + 0.297i)25-s + (−1.05 + 0.611i)29-s + (0.129 − 0.223i)31-s − 1.38i·35-s − 1.63·37-s + (−0.710 − 0.410i)41-s + (0.106 + 0.184i)43-s + (−0.312 + 0.180i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-0.0871 + 0.996i$
Analytic conductor: \(35.3134\)
Root analytic conductor: \(5.94251\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1),\ -0.0871 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.936702052\)
\(L(\frac12)\) \(\approx\) \(1.936702052\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-5.01 - 2.89i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (4.19 + 7.26i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-12.7 + 7.34i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-10.5 + 18.3i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 7.76iT - 289T^{2} \)
19 \( 1 + 24.3T + 361T^{2} \)
23 \( 1 + (-12.7 - 7.34i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (30.7 - 17.7i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 60.5T + 1.36e3T^{2} \)
41 \( 1 + (29.1 + 16.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-4.58 - 7.94i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (14.6 - 8.48i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 25.7iT - 2.80e3T^{2} \)
59 \( 1 + (53.4 + 30.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-10.5 + 18.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 101. iT - 5.04e3T^{2} \)
73 \( 1 - 40.4T + 5.32e3T^{2} \)
79 \( 1 + (-49.3 - 85.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-89.6 + 51.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 134. iT - 7.92e3T^{2} \)
97 \( 1 + (-37.5 - 65.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.293393028383765699756318092103, −8.588452348719352063739895459409, −7.47785480774751116055963939622, −6.56293864774787480000681158670, −6.17963570682298558318715215192, −5.15830411013598623360714414296, −3.71294412411388168812906000977, −3.25240198877513476623852414390, −1.78351862628379756661445617041, −0.53877430931204847784875041503, 1.57608946114708295516263070857, 2.13357193701476544533653830569, 3.67250789649088984529354782896, 4.57336962838026143997528288757, 5.65726412173681820276053714714, 6.37968784467399436046325233018, 6.85171696218346853992835296351, 8.445981461975063277820835230108, 9.120384327261358445947972752155, 9.308831265360477046646708259960

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