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Properties

Label 2-3456-9.4-c1-0-7
Degree $2$
Conductor $3456$
Sign $-0.447 - 0.894i$
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  + (0.268 − 0.464i)5-s + (2.35 + 4.07i)7-s + (−2.59 − 4.50i)11-s + (0.778 − 1.34i)13-s − 0.695·17-s − 5.80·19-s + (−4.42 + 7.66i)23-s + (2.35 + 4.08i)25-s + (1.92 + 3.32i)29-s + (2.77 − 4.79i)31-s + 2.52·35-s − 4.09·37-s + (−1.01 + 1.74i)41-s + (3.71 + 6.43i)43-s + (0.186 + 0.322i)47-s + ⋯
L(s)  = 1  + (0.119 − 0.207i)5-s + (0.888 + 1.53i)7-s + (−0.783 − 1.35i)11-s + (0.215 − 0.373i)13-s − 0.168·17-s − 1.33·19-s + (−0.923 + 1.59i)23-s + (0.471 + 0.816i)25-s + (0.356 + 0.618i)29-s + (0.497 − 0.861i)31-s + 0.426·35-s − 0.672·37-s + (−0.157 + 0.273i)41-s + (0.566 + 0.981i)43-s + (0.0271 + 0.0470i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 3456,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.220207682\)
\(L(\frac12)\) \(\approx\) \(1.220207682\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.268 + 0.464i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.35 - 4.07i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 + 4.50i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.778 + 1.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.695T + 17T^{2} \)
19 \( 1 + 5.80T + 19T^{2} \)
23 \( 1 + (4.42 - 7.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.92 - 3.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.77 + 4.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.09T + 37T^{2} \)
41 \( 1 + (1.01 - 1.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.71 - 6.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.186 - 0.322i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.30T + 53T^{2} \)
59 \( 1 + (2.57 - 4.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.921 - 1.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.79 - 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 4.40T + 73T^{2} \)
79 \( 1 + (-3.32 - 5.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.28 - 9.14i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.30T + 89T^{2} \)
97 \( 1 + (7.81 + 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.683941630457521169501124190191, −8.259587690985835280630147188417, −7.61599113533100594608972161891, −6.33965407486296157432013182381, −5.61946494797952377537333316258, −5.36409363139752957153190044355, −4.28465472025157572057639769266, −3.14048533440828396046806677773, −2.38829219834094687149629851381, −1.36828758952520575103302178667, 0.35225405288284418215998488444, 1.78225917440454406293015615886, 2.47329471683324664023258518206, 3.96955644258983526381198539317, 4.46193954904512514382083858810, 5.00305238156260436784988026386, 6.43564140203521219861282589247, 6.79344164098179204022926225135, 7.67554608712283307264072250175, 8.188132957014641397106140791707

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