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Properties

Label 2-3456-9.7-c1-0-23
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

\(L(1)\) \(\approx\) \(0.8627594346\)
\(L(\frac12)\) \(\approx\) \(0.8627594346\)
\(L(\frac{3}{2})\) 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

−8.599638068653453390972253807897, −7.897286792773374928725772224557, −6.61101858028886282502820020421, −6.44454278573704247147361331513, −5.44215085898542460061601429332, −4.42037328247696961236580258168, −4.04817893476623600843952830526, −2.46189446294982121175366305578, −2.17960120351476311381839466399, −0.31577555682606964237574623359, 0.898384773944383416893129553017, 2.33722959735180337329104812007, 3.44120259032203590247032040372, 3.80121687181926502024593686217, 4.92291761094589407971832039140, 5.80130725874363153389157722722, 6.73955062425335223258594563446, 7.03829387967922348771242184971, 8.039934075112302077567012315100, 8.634703563721749533929019501166

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