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Properties

Label 2-3456-3.2-c0-0-3
Degree $2$
Conductor $3456$
Sign $-i$
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  + 1.41i·5-s + 7-s + 1.41i·11-s + 13-s + 1.41i·17-s − 19-s − 1.41i·23-s − 1.00·25-s + 1.41i·35-s − 37-s − 1.41i·47-s − 2.00·55-s − 1.41i·59-s + 61-s + 1.41i·65-s + ⋯
L(s)  = 1  + 1.41i·5-s + 7-s + 1.41i·11-s + 13-s + 1.41i·17-s − 19-s − 1.41i·23-s − 1.00·25-s + 1.41i·35-s − 37-s − 1.41i·47-s − 2.00·55-s − 1.41i·59-s + 61-s + 1.41i·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.407707402\)
\(L(\frac12)\) \(\approx\) \(1.407707402\)
\(L(1)\) 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 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.638474152078854761338481815823, −8.322055568218347605093372472042, −7.38983042927570819331215210865, −6.66526604375942256225459890568, −6.25080052629596055191561128715, −5.11074436983203014813712743507, −4.23111026118041311325668320002, −3.57764412269110508618615973117, −2.30127087014117305992839593667, −1.76897823987341728681822375025, 0.881251072707869301189100757165, 1.69348437520704524935400958490, 3.08986442919647728751369702306, 4.05975683634840486737955174750, 4.84011980589783955403635270107, 5.48997209770654424317803198869, 6.09359042713328127381031405158, 7.28555267391817167206524553853, 8.053118655834723535928898592630, 8.789051655676995843956165973330

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