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Properties

Label 2-2475-5.4-c1-0-0
Degree $2$
Conductor $2475$
Sign $0.447 + 0.894i$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
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  + 1.90i·2-s − 1.62·4-s + 4.42i·7-s + 0.719i·8-s − 11-s − 0.622i·13-s − 8.42·14-s − 4.61·16-s − 5.18i·17-s − 7.05·19-s − 1.90i·22-s − 8.85i·23-s + 1.18·26-s − 7.18i·28-s − 7.80·29-s + ⋯
L(s)  = 1  + 1.34i·2-s − 0.811·4-s + 1.67i·7-s + 0.254i·8-s − 0.301·11-s − 0.172i·13-s − 2.25·14-s − 1.15·16-s − 1.25i·17-s − 1.61·19-s − 0.405i·22-s − 1.84i·23-s + 0.232·26-s − 1.35i·28-s − 1.44·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2478272938\)
\(L(\frac12)\) \(\approx\) \(0.2478272938\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good2 \( 1 - 1.90iT - 2T^{2} \)
7 \( 1 - 4.42iT - 7T^{2} \)
13 \( 1 + 0.622iT - 13T^{2} \)
17 \( 1 + 5.18iT - 17T^{2} \)
19 \( 1 + 7.05T + 19T^{2} \)
23 \( 1 + 8.85iT - 23T^{2} \)
29 \( 1 + 7.80T + 29T^{2} \)
31 \( 1 - 2.75T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 0.193T + 41T^{2} \)
43 \( 1 - 5.67iT - 43T^{2} \)
47 \( 1 + 2.75iT - 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 + 4.85T + 59T^{2} \)
61 \( 1 - 6.85T + 61T^{2} \)
67 \( 1 - 1.24iT - 67T^{2} \)
71 \( 1 + 2.75T + 71T^{2} \)
73 \( 1 - 4.23iT - 73T^{2} \)
79 \( 1 + 8.56T + 79T^{2} \)
83 \( 1 + 0.133iT - 83T^{2} \)
89 \( 1 - 5.61T + 89T^{2} \)
97 \( 1 + 7.24iT - 97T^{2} \)
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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

−9.044434464481245544043054080816, −8.708954109255271192610161511515, −8.052135556422932264769527930210, −7.19666055898254301071798711879, −6.33809193863908058329415174759, −5.90827274735706739815522853979, −5.07159907913348778936118957880, −4.44071437367594941192106992217, −2.76601422863649332459537230057, −2.21980072028892335002947113820, 0.07793272493196978172763638318, 1.39298710667495261494377066939, 2.12963505941075567247049638619, 3.63026384621231595027956916338, 3.78888817661061832529700081077, 4.70374699561850029766653023903, 5.96301599622954239757438993454, 6.88102829081873141914332729522, 7.54266605097704084001826076658, 8.411721118428434053253878151216

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