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Properties

Label 2-6e4-36.11-c1-0-9
Degree $2$
Conductor $1296$
Sign $0.642 - 0.766i$
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  + (3 + 1.73i)7-s + (1 + 1.73i)13-s + 3.46i·19-s + (−2.5 + 4.33i)25-s + (9 − 5.19i)31-s − 10·37-s + (9 + 5.19i)43-s + (2.5 + 4.33i)49-s + (−7 + 12.1i)61-s + (3 − 1.73i)67-s + 10·73-s + (15 + 8.66i)79-s + 6.92i·91-s + (7 − 12.1i)97-s + (−3 + 1.73i)103-s + ⋯
L(s)  = 1  + (1.13 + 0.654i)7-s + (0.277 + 0.480i)13-s + 0.794i·19-s + (−0.5 + 0.866i)25-s + (1.61 − 0.933i)31-s − 1.64·37-s + (1.37 + 0.792i)43-s + (0.357 + 0.618i)49-s + (−0.896 + 1.55i)61-s + (0.366 − 0.211i)67-s + 1.17·73-s + (1.68 + 0.974i)79-s + 0.726i·91-s + (0.710 − 1.23i)97-s + (−0.295 + 0.170i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.845375430\)
\(L(\frac12)\) \(\approx\) \(1.845375430\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 - 1.73i)T + (3.5 + 6.06i)T^{2} \)
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 \( 1 - 3.46iT - 19T^{2} \)
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 \( 1 + 10T + 37T^{2} \)
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 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3 + 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + (-15 - 8.66i)T + (39.5 + 68.4i)T^{2} \)
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

−9.695102564495582513061001120313, −8.884441370945874751275475248230, −8.161342242555070300837797222769, −7.50270278816317665470934609842, −6.35557129436820882046610984192, −5.56563574328918437073131859404, −4.71016922557058554697422757153, −3.76012528929433354683155657972, −2.41708711245375389993699287221, −1.41358090655469809474270856482, 0.858631287956972339001715510390, 2.15526137665032853602320506405, 3.44405073485393479293543725224, 4.52153552996895230192165566104, 5.14236233033507102880831566595, 6.29033512708065166736983486592, 7.15785338676606816682324884144, 8.013045848557568763734097956860, 8.547247959386689650735290646178, 9.559222035679085571375690140104

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