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Properties

Label 2-3456-9.4-c1-0-19
Degree $2$
Conductor $3456$
Sign $-0.378 - 0.925i$
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  + (−1.05 + 1.82i)5-s + (1.43 + 2.49i)7-s + (1.21 + 2.10i)11-s + (−3.30 + 5.71i)13-s + 7.56·17-s + 6.25·19-s + (2.63 − 4.56i)23-s + (0.275 + 0.476i)25-s + (−1.57 − 2.73i)29-s + (−1.79 + 3.10i)31-s − 6.07·35-s + 6.70·37-s + (1.74 − 3.02i)41-s + (3.12 + 5.41i)43-s + (−1.32 − 2.30i)47-s + ⋯
L(s)  = 1  + (−0.471 + 0.816i)5-s + (0.543 + 0.942i)7-s + (0.365 + 0.633i)11-s + (−0.915 + 1.58i)13-s + 1.83·17-s + 1.43·19-s + (0.549 − 0.952i)23-s + (0.0550 + 0.0953i)25-s + (−0.293 − 0.507i)29-s + (−0.321 + 0.556i)31-s − 1.02·35-s + 1.10·37-s + (0.272 − 0.472i)41-s + (0.476 + 0.825i)43-s + (−0.193 − 0.335i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-0.378 - 0.925i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ -0.378 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.963790315\)
\(L(\frac12)\) \(\approx\) \(1.963790315\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.05 - 1.82i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.43 - 2.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.21 - 2.10i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.30 - 5.71i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.56T + 17T^{2} \)
19 \( 1 - 6.25T + 19T^{2} \)
23 \( 1 + (-2.63 + 4.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.57 + 2.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.79 - 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.70T + 37T^{2} \)
41 \( 1 + (-1.74 + 3.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.12 - 5.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.32 + 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.953T + 53T^{2} \)
59 \( 1 + (4.84 - 8.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.57 + 4.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.949 + 1.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.82T + 71T^{2} \)
73 \( 1 + 5.01T + 73T^{2} \)
79 \( 1 + (-6.49 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.54 + 2.67i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.95T + 89T^{2} \)
97 \( 1 + (5.51 + 9.54i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.955498137182416040376366339251, −7.85505172181129589470521788102, −7.38067070362678570305190466772, −6.76626644798431960510853797734, −5.77691340170261742808704596304, −5.01687660891567700237749032365, −4.23461747452448177533794425458, −3.18908565293190342178486523354, −2.43606222723455238813327281078, −1.36485932069361283003644076658, 0.72859632369510035603138775459, 1.20455818701628626047009935011, 3.01289337901544737854382823645, 3.56653388738693434199756154582, 4.60037578372665765621627121297, 5.34045870767753726666791893633, 5.79783127964977826766764576611, 7.28300873618967222342874025029, 7.73284615055838909334808764854, 8.021648679141662122179441207415

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