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Properties

Label 2-3456-9.7-c1-0-19
Degree $2$
Conductor $3456$
Sign $0.966 - 0.257i$
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  + (0.115 + 0.200i)5-s + (−0.230 + 0.399i)7-s + (0.749 − 1.29i)11-s + (1.07 + 1.85i)13-s − 1.03·17-s − 2.94·19-s + (0.364 + 0.631i)23-s + (2.47 − 4.28i)25-s + (2.33 − 4.04i)29-s + (2.73 + 4.73i)31-s − 0.106·35-s + 2.30·37-s + (1.84 + 3.18i)41-s + (−2.41 + 4.19i)43-s + (5.40 − 9.36i)47-s + ⋯
L(s)  = 1  + (0.0518 + 0.0897i)5-s + (−0.0872 + 0.151i)7-s + (0.225 − 0.391i)11-s + (0.296 + 0.514i)13-s − 0.251·17-s − 0.675·19-s + (0.0760 + 0.131i)23-s + (0.494 − 0.856i)25-s + (0.433 − 0.751i)29-s + (0.491 + 0.851i)31-s − 0.0180·35-s + 0.378·37-s + (0.287 + 0.498i)41-s + (−0.368 + 0.639i)43-s + (0.788 − 1.36i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.966 - 0.257i$
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: \((2,\ 3456,\ (\ :1/2),\ 0.966 - 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.846199265\)
\(L(\frac12)\) \(\approx\) \(1.846199265\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.115 - 0.200i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.230 - 0.399i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.749 + 1.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.07 - 1.85i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.03T + 17T^{2} \)
19 \( 1 + 2.94T + 19T^{2} \)
23 \( 1 + (-0.364 - 0.631i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.33 + 4.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.73 - 4.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.30T + 37T^{2} \)
41 \( 1 + (-1.84 - 3.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.41 - 4.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.40 + 9.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + (2.71 + 4.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.86 - 11.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.58 + 9.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 + (6.23 - 10.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.62 - 6.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (2.21 - 3.83i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.620669539486071159507623920285, −8.032929502959884575850312337202, −6.98051782792951382149746495216, −6.41570647107080231755547333078, −5.74888709076670750651762119690, −4.67827171091752741123850156249, −4.05290358924092702160226549008, −2.99749880964965453880567272187, −2.13830346693095069206408848416, −0.862969514227233423734292324636, 0.76510681221874101774896505961, 1.97716612761046715272879909968, 2.98571017555737104319392089864, 3.93685471858548392826549511226, 4.69178671047104870375717645167, 5.57047783599107327069868713888, 6.33827203041353777626104022494, 7.09831473712334189889532306012, 7.77379760025004275911316833986, 8.680634650393388856019979536084

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