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Properties

Label 2-15e3-15.14-c0-0-7
Degree $2$
Conductor $3375$
Sign $1$
Analytic cond. $1.68434$
Root an. cond. $1.29782$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.95·2-s + 2.82·4-s + 3.57·8-s + 4.16·16-s + 0.209·17-s − 1.95·19-s − 1.82·23-s − 0.209·31-s + 4.57·32-s + 0.408·34-s − 3.82·38-s − 3.57·46-s − 0.618·47-s + 49-s − 1.33·53-s + 1.33·61-s − 0.408·62-s + 4.78·64-s + 0.591·68-s − 5.53·76-s + 1.33·79-s − 1.33·83-s − 5.16·92-s − 1.20·94-s + 1.95·98-s − 2.61·106-s + 1.61·107-s + ⋯
L(s)  = 1  + 1.95·2-s + 2.82·4-s + 3.57·8-s + 4.16·16-s + 0.209·17-s − 1.95·19-s − 1.82·23-s − 0.209·31-s + 4.57·32-s + 0.408·34-s − 3.82·38-s − 3.57·46-s − 0.618·47-s + 49-s − 1.33·53-s + 1.33·61-s − 0.408·62-s + 4.78·64-s + 0.591·68-s − 5.53·76-s + 1.33·79-s − 1.33·83-s − 5.16·92-s − 1.20·94-s + 1.95·98-s − 2.61·106-s + 1.61·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3375\)    =    \(3^{3} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(1.68434\)
Root analytic conductor: \(1.29782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3375} (3374, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3375,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.254442892\)
\(L(\frac12)\) \(\approx\) \(4.254442892\)
\(L(1)\) 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 \( 1 \)
good2 \( 1 - 1.95T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 0.209T + T^{2} \)
19 \( 1 + 1.95T + T^{2} \)
23 \( 1 + 1.82T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 0.209T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 0.618T + T^{2} \)
53 \( 1 + 1.33T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.33T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.33T + T^{2} \)
83 \( 1 + 1.33T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.477304350642184927681497986053, −7.82511674719415767191483620629, −6.94012735745981036041085708913, −6.25810084853121976880533672972, −5.78482834826609647287681435828, −4.84010186163237370841559412812, −4.16129591324722976896222386764, −3.57565393631589973761530901516, −2.46649009967079108613250911531, −1.81731124387847037569105103512, 1.81731124387847037569105103512, 2.46649009967079108613250911531, 3.57565393631589973761530901516, 4.16129591324722976896222386764, 4.84010186163237370841559412812, 5.78482834826609647287681435828, 6.25810084853121976880533672972, 6.94012735745981036041085708913, 7.82511674719415767191483620629, 8.477304350642184927681497986053

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