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Properties

Label 2-3456-9.7-c1-0-9
Degree $2$
Conductor $3456$
Sign $-0.999 - 0.0334i$
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  + (1.34 + 2.32i)5-s + (−2.48 + 4.30i)7-s + (−1.26 + 2.19i)11-s + (2.21 + 3.83i)13-s + 2.43·17-s − 4.18·19-s + (−0.570 − 0.988i)23-s + (−1.09 + 1.89i)25-s + (−3.00 + 5.20i)29-s + (2.65 + 4.59i)31-s − 13.3·35-s + 0.241·37-s + (3.21 + 5.56i)41-s + (5.57 − 9.65i)43-s + (−2.37 + 4.11i)47-s + ⋯
L(s)  = 1  + (0.599 + 1.03i)5-s + (−0.939 + 1.62i)7-s + (−0.382 + 0.662i)11-s + (0.614 + 1.06i)13-s + 0.590·17-s − 0.959·19-s + (−0.118 − 0.206i)23-s + (−0.218 + 0.378i)25-s + (−0.557 + 0.966i)29-s + (0.476 + 0.825i)31-s − 2.25·35-s + 0.0397·37-s + (0.501 + 0.868i)41-s + (0.849 − 1.47i)43-s + (−0.346 + 0.600i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-0.999 - 0.0334i$
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: \((2,\ 3456,\ (\ :1/2),\ -0.999 - 0.0334i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.453133400\)
\(L(\frac12)\) \(\approx\) \(1.453133400\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.34 - 2.32i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.48 - 4.30i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.26 - 2.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.21 - 3.83i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.43T + 17T^{2} \)
19 \( 1 + 4.18T + 19T^{2} \)
23 \( 1 + (0.570 + 0.988i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.00 - 5.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.65 - 4.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.241T + 37T^{2} \)
41 \( 1 + (-3.21 - 5.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.57 + 9.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.37 - 4.11i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.38T + 53T^{2} \)
59 \( 1 + (-5.40 - 9.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.16 + 7.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.13 - 1.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.52T + 71T^{2} \)
73 \( 1 + 3.34T + 73T^{2} \)
79 \( 1 + (-3.14 + 5.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.738 - 1.27i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + (-5.89 + 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.913956078023218515698591370773, −8.496871910698020016917574703959, −7.17680896670560637974444374287, −6.68356601060436952202413727245, −5.99866087365140751527403964255, −5.47360668358183467300806530007, −4.30018193100744113334277914858, −3.20917940555770726493673182855, −2.52844170174297219593243813056, −1.83997607353659420077098092693, 0.47254312239284009277023953140, 1.10947076296774432843422348989, 2.59095784602464888710664035274, 3.68710986161098088194925751794, 4.17507589432674855773294644202, 5.32438908952576238063058799881, 5.92614799777384386599403623029, 6.61801846949470182082221758331, 7.68757968544273321636572429030, 8.097842189278212289005614683782

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