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Properties

Label 2-3456-9.4-c1-0-12
Degree $2$
Conductor $3456$
Sign $0.408 - 0.912i$
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  + (−1.07 + 1.86i)5-s + (0.153 + 0.265i)7-s + (−2.50 − 4.34i)11-s + (0.470 − 0.815i)13-s + 4.70·17-s + 1.61·19-s + (−4.08 + 7.06i)23-s + (0.191 + 0.330i)25-s + (−2.39 − 4.14i)29-s + (−1.29 + 2.24i)31-s − 0.658·35-s + 10.2·37-s + (−3.86 + 6.69i)41-s + (0.138 + 0.239i)43-s + (−1.92 − 3.32i)47-s + ⋯
L(s)  = 1  + (−0.480 + 0.832i)5-s + (0.0578 + 0.100i)7-s + (−0.755 − 1.30i)11-s + (0.130 − 0.226i)13-s + 1.14·17-s + 0.371·19-s + (−0.851 + 1.47i)23-s + (0.0382 + 0.0661i)25-s + (−0.444 − 0.770i)29-s + (−0.233 + 0.403i)31-s − 0.111·35-s + 1.67·37-s + (−0.603 + 1.04i)41-s + (0.0210 + 0.0364i)43-s + (−0.280 − 0.485i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.408 - 0.912i$
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: \((2,\ 3456,\ (\ :1/2),\ 0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.413495806\)
\(L(\frac12)\) \(\approx\) \(1.413495806\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.07 - 1.86i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.153 - 0.265i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.50 + 4.34i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.470 + 0.815i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.70T + 17T^{2} \)
19 \( 1 - 1.61T + 19T^{2} \)
23 \( 1 + (4.08 - 7.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.39 + 4.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.29 - 2.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + (3.86 - 6.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.138 - 0.239i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.92 + 3.32i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.23T + 53T^{2} \)
59 \( 1 + (-4.95 + 8.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.36 - 9.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.02 + 3.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.59T + 71T^{2} \)
73 \( 1 + 5.43T + 73T^{2} \)
79 \( 1 + (-8.30 - 14.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.91 + 5.05i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.94T + 89T^{2} \)
97 \( 1 + (-7.07 - 12.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.514874716071140488461436136340, −7.83822583314692662571001198599, −7.51390778433900986221955299291, −6.41917567961922633857909136138, −5.71346030973082693999804228647, −5.13308718983491245044277651154, −3.66575832467713901209892195294, −3.39063237157834943195077828040, −2.40253459749084285899775691265, −0.944716057531698841123433432041, 0.53459329385605226662442417661, 1.77733607082383070263331318432, 2.79351030470617812996378103242, 3.99954933511706284347583137486, 4.58257242853228297603029806020, 5.28685934603947196432303695222, 6.14950208258916122840761331458, 7.19837293643991805126773131968, 7.73950567788722653221116606361, 8.356143512504077137882633139214

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