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Properties

Label 2-6e4-36.23-c1-0-5
Degree $2$
Conductor $1296$
Sign $0.342 - 0.939i$
Analytic cond. $10.3486$
Root an. cond. $3.21692$
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  + (1.5 − 0.866i)7-s + (−3.5 + 6.06i)13-s + 8.66i·19-s + (−2.5 − 4.33i)25-s + (9 + 5.19i)31-s − 37-s + (9 − 5.19i)43-s + (−2 + 3.46i)49-s + (6.5 + 11.2i)61-s + (10.5 + 6.06i)67-s − 17·73-s + (−10.5 + 6.06i)79-s + 12.1i·91-s + (2.5 + 4.33i)97-s + (16.5 + 9.52i)103-s + ⋯
L(s)  = 1  + (0.566 − 0.327i)7-s + (−0.970 + 1.68i)13-s + 1.98i·19-s + (−0.5 − 0.866i)25-s + (1.61 + 0.933i)31-s − 0.164·37-s + (1.37 − 0.792i)43-s + (−0.285 + 0.494i)49-s + (0.832 + 1.44i)61-s + (1.28 + 0.740i)67-s − 1.98·73-s + (−1.18 + 0.682i)79-s + 1.27i·91-s + (0.253 + 0.439i)97-s + (1.62 + 0.938i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.464675449\)
\(L(\frac12)\) \(\approx\) \(1.464675449\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 + 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.5 - 6.06i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 8.66iT - 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9 - 5.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9 + 5.19i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.5 - 6.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 17T + 73T^{2} \)
79 \( 1 + (10.5 - 6.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.987931261664538455981097207418, −8.961323336091691277515780311441, −8.175405868009276964261154139925, −7.39027513940284483757819696950, −6.58092812848564055919857418870, −5.64841656381367858398908003175, −4.52279972003395149616113414066, −3.99183490254623687266768140380, −2.48779330918976531059836383661, −1.44303450871384014936075293059, 0.63325911788407493998414587520, 2.33102846419079913114249667782, 3.11839595404436634647512563178, 4.57676811708720689916874418388, 5.18188997729255598737337648950, 6.08088428300557787770974127448, 7.21945113043811537508773283024, 7.84032270905935207785416255930, 8.637402297087021235465809745787, 9.557008989735424653239844180729

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