Properties

Label 2-1875-1.1-c3-0-210
Degree $2$
Conductor $1875$
Sign $-1$
Analytic cond. $110.628$
Root an. cond. $10.5180$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.93·2-s + 3·3-s + 0.630·4-s − 8.81·6-s + 18.9·7-s + 21.6·8-s + 9·9-s − 6.06·11-s + 1.89·12-s + 78.2·13-s − 55.5·14-s − 68.6·16-s + 38.4·17-s − 26.4·18-s − 92.9·19-s + 56.7·21-s + 17.8·22-s − 81.0·23-s + 64.9·24-s − 229.·26-s + 27·27-s + 11.9·28-s − 11.1·29-s − 223.·31-s + 28.4·32-s − 18.2·33-s − 112.·34-s + ⋯
L(s)  = 1  − 1.03·2-s + 0.577·3-s + 0.0788·4-s − 0.599·6-s + 1.02·7-s + 0.956·8-s + 0.333·9-s − 0.166·11-s + 0.0455·12-s + 1.66·13-s − 1.06·14-s − 1.07·16-s + 0.547·17-s − 0.346·18-s − 1.12·19-s + 0.589·21-s + 0.172·22-s − 0.734·23-s + 0.552·24-s − 1.73·26-s + 0.192·27-s + 0.0805·28-s − 0.0714·29-s − 1.29·31-s + 0.157·32-s − 0.0960·33-s − 0.569·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(110.628\)
Root analytic conductor: \(10.5180\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1875,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 \)
good2 \( 1 + 2.93T + 8T^{2} \)
7 \( 1 - 18.9T + 343T^{2} \)
11 \( 1 + 6.06T + 1.33e3T^{2} \)
13 \( 1 - 78.2T + 2.19e3T^{2} \)
17 \( 1 - 38.4T + 4.91e3T^{2} \)
19 \( 1 + 92.9T + 6.85e3T^{2} \)
23 \( 1 + 81.0T + 1.21e4T^{2} \)
29 \( 1 + 11.1T + 2.43e4T^{2} \)
31 \( 1 + 223.T + 2.97e4T^{2} \)
37 \( 1 + 107.T + 5.06e4T^{2} \)
41 \( 1 + 350.T + 6.89e4T^{2} \)
43 \( 1 + 356.T + 7.95e4T^{2} \)
47 \( 1 + 291.T + 1.03e5T^{2} \)
53 \( 1 - 684.T + 1.48e5T^{2} \)
59 \( 1 + 223.T + 2.05e5T^{2} \)
61 \( 1 + 697.T + 2.26e5T^{2} \)
67 \( 1 + 909.T + 3.00e5T^{2} \)
71 \( 1 - 201.T + 3.57e5T^{2} \)
73 \( 1 + 291.T + 3.89e5T^{2} \)
79 \( 1 - 709.T + 4.93e5T^{2} \)
83 \( 1 + 1.10e3T + 5.71e5T^{2} \)
89 \( 1 - 3.77T + 7.04e5T^{2} \)
97 \( 1 - 18.4T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.552524288402296319920664915730, −8.036344299624704961188254129486, −7.28512774116924520580698225083, −6.24817942923156069020841561334, −5.17173263570908966620922184695, −4.24233438986693811896093897335, −3.43790659197265624820137947886, −1.86023140571303293097269545125, −1.42202897673161948304178206609, 0, 1.42202897673161948304178206609, 1.86023140571303293097269545125, 3.43790659197265624820137947886, 4.24233438986693811896093897335, 5.17173263570908966620922184695, 6.24817942923156069020841561334, 7.28512774116924520580698225083, 8.036344299624704961188254129486, 8.552524288402296319920664915730

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