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Properties

Label 2-3456-9.4-c1-0-37
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: $\chi_{3456} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ -0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.005814014\)
\(L(\frac12)\) \(\approx\) \(1.005814014\)
\(L(\frac{3}{2})\) 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

−8.456354527750079135800169210367, −7.86654776387443788493959075986, −6.64882424924423979522680795556, −6.20426885778104938632932066414, −5.16841353960717103248346390650, −4.74615382073383901041028478986, −3.73974446016147483089003403631, −2.31203125442949669216149953492, −2.05929058855949309382387085993, −0.29028805930581955758133725533, 1.28031163045853728141050591685, 2.33691588866699643273669846953, 3.23562475558320273873045827089, 4.44437777039286673259566151925, 4.75499266318035070692956225196, 5.84360326139086773311634789559, 6.84611849538332262497634899304, 7.18009533312675037730099305420, 8.049287652490182269901913272719, 8.686606112063547669189051001876

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