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Properties

Label 2-3456-72.5-c0-0-0
Degree $2$
Conductor $3456$
Sign $0.766 - 0.642i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)11-s + 1.73i·17-s + 1.73i·19-s + (0.5 + 0.866i)25-s + (1.5 + 0.866i)41-s + (1.5 − 0.866i)43-s + (0.5 − 0.866i)49-s + (0.5 − 0.866i)59-s + (−1.5 − 0.866i)67-s + 73-s + (1 + 1.73i)83-s + (0.5 + 0.866i)97-s − 107-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)11-s + 1.73i·17-s + 1.73i·19-s + (0.5 + 0.866i)25-s + (1.5 + 0.866i)41-s + (1.5 − 0.866i)43-s + (0.5 − 0.866i)49-s + (0.5 − 0.866i)59-s + (−1.5 − 0.866i)67-s + 73-s + (1 + 1.73i)83-s + (0.5 + 0.866i)97-s − 107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :0),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.189262134\)
\(L(\frac12)\) \(\approx\) \(1.189262134\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 1.73iT - T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.756243255270413277445148560963, −8.057792873275272641467578760481, −7.61075595217461871550395065141, −6.42842668215301993280302495883, −5.88810212681014885249842779245, −5.23202433580605034945114044695, −3.99494442585858812411607632281, −3.52373042053368596623918984989, −2.34849937111716052995174171509, −1.25709237993783723197365317116, 0.78579366576401502264872229849, 2.46353074248081841693638434779, 2.79612690409061597786414653360, 4.30785127641375622202987455605, 4.75775723253934762740851963097, 5.59215017639522523449782434984, 6.58200879885822621157896248902, 7.34130410061758676422642922417, 7.67510222677835198064419812421, 9.041182503054297003179263747125

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