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Properties

Label 2-3456-9.7-c1-0-33
Degree $2$
Conductor $3456$
Sign $0.336 + 0.941i$
Analytic cond. $27.5962$
Root an. cond. $5.25321$
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  + (−0.551 − 0.955i)5-s + (1.62 − 2.81i)7-s + (1.28 − 2.23i)11-s + (1.58 + 2.74i)13-s − 4.71·17-s + 5.75·19-s + (2.35 + 4.07i)23-s + (1.89 − 3.27i)25-s + (3.66 − 6.34i)29-s + (2.93 + 5.07i)31-s − 3.58·35-s − 0.0714·37-s + (1.63 + 2.83i)41-s + (2.12 − 3.67i)43-s + (−4.72 + 8.18i)47-s + ⋯
L(s)  = 1  + (−0.246 − 0.427i)5-s + (0.614 − 1.06i)7-s + (0.388 − 0.672i)11-s + (0.440 + 0.762i)13-s − 1.14·17-s + 1.32·19-s + (0.490 + 0.850i)23-s + (0.378 − 0.655i)25-s + (0.680 − 1.17i)29-s + (0.526 + 0.911i)31-s − 0.605·35-s − 0.0117·37-s + (0.255 + 0.443i)41-s + (0.323 − 0.560i)43-s + (−0.689 + 1.19i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ 0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.034452090\)
\(L(\frac12)\) \(\approx\) \(2.034452090\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.551 + 0.955i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.62 + 2.81i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.28 + 2.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.58 - 2.74i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.71T + 17T^{2} \)
19 \( 1 - 5.75T + 19T^{2} \)
23 \( 1 + (-2.35 - 4.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.66 + 6.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.93 - 5.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.0714T + 37T^{2} \)
41 \( 1 + (-1.63 - 2.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.12 + 3.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.72 - 8.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.42T + 53T^{2} \)
59 \( 1 + (4.19 + 7.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.66 + 8.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.09 + 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.335T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 + (4.85 - 8.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.07 - 5.31i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.42T + 89T^{2} \)
97 \( 1 + (-6.39 + 11.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.336344739471763323458653396741, −7.84489751291984400185822728851, −6.89587747195292940515580403310, −6.39682142503710754182178178626, −5.26105519497771688099900727004, −4.49839623909350267063150048204, −3.94501523334437874812105414637, −2.93740367887805887175710868924, −1.54072939320223738343274220104, −0.73530372106753725319480754724, 1.14701699012224939077150000775, 2.35704294407173698052748501518, 3.04376402224758983086228659781, 4.13628812050859251699257667584, 5.02917572075886700053364767482, 5.61156783463601897765813299987, 6.61644807192836172819469357390, 7.19646165100694964378355640214, 8.052360147249317240389679664058, 8.795899762300219589894063121075

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