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Properties

Label 2-6e4-9.4-c1-0-22
Degree $2$
Conductor $1296$
Sign $-0.939 - 0.342i$
Analytic cond. $10.3486$
Root an. cond. $3.21692$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)7-s + (−1 + 1.73i)13-s − 8·19-s + (2.5 + 4.33i)25-s + (−2 + 3.46i)31-s − 10·37-s + (4 + 6.92i)43-s + (−4.49 + 7.79i)49-s + (−7 − 12.1i)61-s + (−8 + 13.8i)67-s − 10·73-s + (−2 − 3.46i)79-s + 7.99·91-s + (−7 − 12.1i)97-s + (10 − 17.3i)103-s + ⋯
L(s)  = 1  + (−0.755 − 1.30i)7-s + (−0.277 + 0.480i)13-s − 1.83·19-s + (0.5 + 0.866i)25-s + (−0.359 + 0.622i)31-s − 1.64·37-s + (0.609 + 1.05i)43-s + (−0.642 + 1.11i)49-s + (−0.896 − 1.55i)61-s + (−0.977 + 1.69i)67-s − 1.17·73-s + (−0.225 − 0.389i)79-s + 0.838·91-s + (−0.710 − 1.23i)97-s + (0.985 − 1.70i)103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8 - 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.211713298004280934576305971546, −8.483252289245808846100882759500, −7.33345691721823336896265067753, −6.86841691726000487303562102600, −6.03614083244652445333918166232, −4.75215650835674560577797473830, −4.00417543536580328788535595394, −3.06771075596467426674622927791, −1.61736177436991935904221204202, 0, 2.09217505903120938420458680056, 2.89245730339382694666960367400, 4.06553442717947310899147176976, 5.17631427365747618133931549080, 6.02312489529303817182564551620, 6.63272656572301314153677296946, 7.73338015764669673333066923383, 8.802252600257081365423287630570, 8.990890859707656633183436792477

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