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Properties

Label 2-1824-456.107-c0-0-1
Degree $2$
Conductor $1824$
Sign $-0.0977 + 0.995i$
Analytic cond. $0.910294$
Root an. cond. $0.954093$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)9-s − 1.73i·11-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)25-s − 0.999·27-s + (−1.49 − 0.866i)33-s + (0.5 + 0.866i)41-s + (−1 − 1.73i)43-s + 49-s + (0.499 + 0.866i)57-s + (0.5 + 0.866i)59-s + (1.5 + 0.866i)67-s + (−0.5 − 0.866i)73-s + (−0.499 − 0.866i)75-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)9-s − 1.73i·11-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)25-s − 0.999·27-s + (−1.49 − 0.866i)33-s + (0.5 + 0.866i)41-s + (−1 − 1.73i)43-s + 49-s + (0.499 + 0.866i)57-s + (0.5 + 0.866i)59-s + (1.5 + 0.866i)67-s + (−0.5 − 0.866i)73-s + (−0.499 − 0.866i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1824\)    =    \(2^{5} \cdot 3 \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1824} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1824,\ (\ :0),\ -0.0977 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.251318916\)
\(L(\frac12)\) \(\approx\) \(1.251318916\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - 1.73iT - T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.848391795990279290429726417054, −8.507527960404828479909677025697, −7.81609023756731912786921364390, −6.83488123569949152808659891811, −6.13938727317749822545592021364, −5.45973250995101557653821515948, −4.00890911363620267570003272316, −3.21087172364731855190408569435, −2.24741594815423850890790132107, −0.903763983401614284097119657853, 1.90843106216821884702046095271, 2.83386487098021772162606038074, 3.95285800793406209613535748281, 4.69760736040101314655235262386, 5.31790153817865953548298865192, 6.62121622685357017176945508634, 7.35763232474525984024877821048, 8.169695131621873464890440710820, 9.093302798590917895492902984088, 9.564428256479058022685684434999

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