Properties

Label 2-1104-1.1-c5-0-106
Degree $2$
Conductor $1104$
Sign $-1$
Analytic cond. $177.063$
Root an. cond. $13.3065$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s + 70.5·5-s + 89.7·7-s + 81·9-s + 436.·11-s − 258.·13-s − 634.·15-s + 591.·17-s − 1.66e3·19-s − 807.·21-s + 529·23-s + 1.84e3·25-s − 729·27-s − 5.12e3·29-s − 1.87e3·31-s − 3.93e3·33-s + 6.32e3·35-s + 884.·37-s + 2.32e3·39-s − 1.84e4·41-s − 1.23e4·43-s + 5.71e3·45-s − 2.38e4·47-s − 8.75e3·49-s − 5.32e3·51-s − 1.45e4·53-s + 3.08e4·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.26·5-s + 0.692·7-s + 0.333·9-s + 1.08·11-s − 0.424·13-s − 0.728·15-s + 0.496·17-s − 1.05·19-s − 0.399·21-s + 0.208·23-s + 0.591·25-s − 0.192·27-s − 1.13·29-s − 0.349·31-s − 0.628·33-s + 0.872·35-s + 0.106·37-s + 0.245·39-s − 1.71·41-s − 1.01·43-s + 0.420·45-s − 1.57·47-s − 0.521·49-s − 0.286·51-s − 0.713·53-s + 1.37·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $-1$
Analytic conductor: \(177.063\)
Root analytic conductor: \(13.3065\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1104,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 9T \)
23 \( 1 - 529T \)
good5 \( 1 - 70.5T + 3.12e3T^{2} \)
7 \( 1 - 89.7T + 1.68e4T^{2} \)
11 \( 1 - 436.T + 1.61e5T^{2} \)
13 \( 1 + 258.T + 3.71e5T^{2} \)
17 \( 1 - 591.T + 1.41e6T^{2} \)
19 \( 1 + 1.66e3T + 2.47e6T^{2} \)
29 \( 1 + 5.12e3T + 2.05e7T^{2} \)
31 \( 1 + 1.87e3T + 2.86e7T^{2} \)
37 \( 1 - 884.T + 6.93e7T^{2} \)
41 \( 1 + 1.84e4T + 1.15e8T^{2} \)
43 \( 1 + 1.23e4T + 1.47e8T^{2} \)
47 \( 1 + 2.38e4T + 2.29e8T^{2} \)
53 \( 1 + 1.45e4T + 4.18e8T^{2} \)
59 \( 1 + 4.77e4T + 7.14e8T^{2} \)
61 \( 1 + 1.53e4T + 8.44e8T^{2} \)
67 \( 1 - 4.11e4T + 1.35e9T^{2} \)
71 \( 1 - 4.56e4T + 1.80e9T^{2} \)
73 \( 1 + 7.89e3T + 2.07e9T^{2} \)
79 \( 1 + 7.14e4T + 3.07e9T^{2} \)
83 \( 1 + 1.12e5T + 3.93e9T^{2} \)
89 \( 1 - 1.37e5T + 5.58e9T^{2} \)
97 \( 1 - 4.16e4T + 8.58e9T^{2} \)
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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

−8.865328093407174592327589275757, −7.88231937940335310331519033754, −6.76338190716568412886845633144, −6.21175826036106926213711846265, −5.30681386278048989056622452202, −4.61763510717310331214805530759, −3.39996929106466424285744594243, −1.88352068951474527052336168166, −1.50611512828374710770749814992, 0, 1.50611512828374710770749814992, 1.88352068951474527052336168166, 3.39996929106466424285744594243, 4.61763510717310331214805530759, 5.30681386278048989056622452202, 6.21175826036106926213711846265, 6.76338190716568412886845633144, 7.88231937940335310331519033754, 8.865328093407174592327589275757

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