Properties

Label 2-1875-1.1-c3-0-235
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.90·2-s + 3·3-s + 7.26·4-s + 11.7·6-s + 22.0·7-s − 2.87·8-s + 9·9-s − 39.0·11-s + 21.7·12-s − 68.3·13-s + 86.3·14-s − 69.3·16-s + 100.·17-s + 35.1·18-s − 94.0·19-s + 66.2·21-s − 152.·22-s − 161.·23-s − 8.63·24-s − 267.·26-s + 27·27-s + 160.·28-s − 51.2·29-s − 260.·31-s − 247.·32-s − 117.·33-s + 394.·34-s + ⋯
L(s)  = 1  + 1.38·2-s + 0.577·3-s + 0.907·4-s + 0.797·6-s + 1.19·7-s − 0.127·8-s + 0.333·9-s − 1.07·11-s + 0.524·12-s − 1.45·13-s + 1.64·14-s − 1.08·16-s + 1.43·17-s + 0.460·18-s − 1.13·19-s + 0.688·21-s − 1.47·22-s − 1.46·23-s − 0.0734·24-s − 2.01·26-s + 0.192·27-s + 1.08·28-s − 0.328·29-s − 1.50·31-s − 1.36·32-s − 0.618·33-s + 1.98·34-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 \( 1 - 3.90T + 8T^{2} \)
7 \( 1 - 22.0T + 343T^{2} \)
11 \( 1 + 39.0T + 1.33e3T^{2} \)
13 \( 1 + 68.3T + 2.19e3T^{2} \)
17 \( 1 - 100.T + 4.91e3T^{2} \)
19 \( 1 + 94.0T + 6.85e3T^{2} \)
23 \( 1 + 161.T + 1.21e4T^{2} \)
29 \( 1 + 51.2T + 2.43e4T^{2} \)
31 \( 1 + 260.T + 2.97e4T^{2} \)
37 \( 1 + 86.6T + 5.06e4T^{2} \)
41 \( 1 + 52.5T + 6.89e4T^{2} \)
43 \( 1 + 53.3T + 7.95e4T^{2} \)
47 \( 1 - 241.T + 1.03e5T^{2} \)
53 \( 1 - 59.1T + 1.48e5T^{2} \)
59 \( 1 + 648.T + 2.05e5T^{2} \)
61 \( 1 - 655.T + 2.26e5T^{2} \)
67 \( 1 - 778.T + 3.00e5T^{2} \)
71 \( 1 - 224.T + 3.57e5T^{2} \)
73 \( 1 - 348.T + 3.89e5T^{2} \)
79 \( 1 - 161.T + 4.93e5T^{2} \)
83 \( 1 + 39.8T + 5.71e5T^{2} \)
89 \( 1 + 463.T + 7.04e5T^{2} \)
97 \( 1 - 574.T + 9.12e5T^{2} \)
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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

−8.132941004539225210822397889856, −7.77984650525961493810241226462, −6.86470401797366011631454483207, −5.57450980718598259595268840535, −5.22689354083239137812301527440, −4.37331160042583114957957423724, −3.59895832164155971650702882079, −2.49917853992084527121904364546, −1.91502751001159855911024172179, 0, 1.91502751001159855911024172179, 2.49917853992084527121904364546, 3.59895832164155971650702882079, 4.37331160042583114957957423724, 5.22689354083239137812301527440, 5.57450980718598259595268840535, 6.86470401797366011631454483207, 7.77984650525961493810241226462, 8.132941004539225210822397889856

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