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Properties

Label 2-825-5.4-c3-0-49
Degree $2$
Conductor $825$
Sign $0.894 - 0.447i$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.26i·2-s − 3i·3-s − 19.6·4-s + 15.7·6-s + 10.3i·7-s − 61.4i·8-s − 9·9-s + 11·11-s + 59.0i·12-s + 63.9i·13-s − 54.3·14-s + 165.·16-s − 17.1i·17-s − 47.3i·18-s − 90.2·19-s + ⋯
L(s)  = 1  + 1.86i·2-s − 0.577i·3-s − 2.46·4-s + 1.07·6-s + 0.557i·7-s − 2.71i·8-s − 0.333·9-s + 0.301·11-s + 1.42i·12-s + 1.36i·13-s − 1.03·14-s + 2.59·16-s − 0.244i·17-s − 0.620i·18-s − 1.08·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8683758708\)
\(L(\frac12)\) \(\approx\) \(0.8683758708\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 - 5.26iT - 8T^{2} \)
7 \( 1 - 10.3iT - 343T^{2} \)
13 \( 1 - 63.9iT - 2.19e3T^{2} \)
17 \( 1 + 17.1iT - 4.91e3T^{2} \)
19 \( 1 + 90.2T + 6.85e3T^{2} \)
23 \( 1 + 212. iT - 1.21e4T^{2} \)
29 \( 1 + 57.5T + 2.43e4T^{2} \)
31 \( 1 + 141.T + 2.97e4T^{2} \)
37 \( 1 - 257. iT - 5.06e4T^{2} \)
41 \( 1 + 225.T + 6.89e4T^{2} \)
43 \( 1 + 347. iT - 7.95e4T^{2} \)
47 \( 1 + 404. iT - 1.03e5T^{2} \)
53 \( 1 - 259. iT - 1.48e5T^{2} \)
59 \( 1 - 853.T + 2.05e5T^{2} \)
61 \( 1 + 203.T + 2.26e5T^{2} \)
67 \( 1 + 266. iT - 3.00e5T^{2} \)
71 \( 1 - 92.4T + 3.57e5T^{2} \)
73 \( 1 + 242. iT - 3.89e5T^{2} \)
79 \( 1 - 1.02e3T + 4.93e5T^{2} \)
83 \( 1 - 706. iT - 5.71e5T^{2} \)
89 \( 1 - 440.T + 7.04e5T^{2} \)
97 \( 1 - 197. iT - 9.12e5T^{2} \)
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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

−9.317123083981973370888742840184, −8.707827551874752072183790133762, −8.204613395578948146656365236933, −6.85270693995958260047025808178, −6.78557383137548751945948070641, −5.80093538313168097969524945206, −4.83794198636038342370295609915, −3.94590424797836244282359280031, −2.10591576118162003286585364675, −0.29333210226067697247864627490, 0.920839449375790964590036914452, 2.13871012016743284835804207660, 3.39635407773799364715554303795, 3.85323246458169940572605926418, 4.93211562044559100897948968175, 5.82232252509367989238130665924, 7.53780469616480412008211462934, 8.487161080627044418988778611796, 9.353832691099224331669861002334, 9.987803004276853830135248231135

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