Properties

Label 2-1875-1.1-c1-0-37
Degree $2$
Conductor $1875$
Sign $1$
Analytic cond. $14.9719$
Root an. cond. $3.86936$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.31·2-s + 3-s − 0.274·4-s + 1.31·6-s + 4.19·7-s − 2.98·8-s + 9-s − 0.167·11-s − 0.274·12-s − 3.39·13-s + 5.50·14-s − 3.37·16-s + 4.57·17-s + 1.31·18-s + 3.78·19-s + 4.19·21-s − 0.220·22-s + 8.31·23-s − 2.98·24-s − 4.45·26-s + 27-s − 1.15·28-s + 2.74·29-s − 7.71·31-s + 1.54·32-s − 0.167·33-s + 6.00·34-s + ⋯
L(s)  = 1  + 0.928·2-s + 0.577·3-s − 0.137·4-s + 0.536·6-s + 1.58·7-s − 1.05·8-s + 0.333·9-s − 0.0505·11-s − 0.0792·12-s − 0.941·13-s + 1.47·14-s − 0.843·16-s + 1.10·17-s + 0.309·18-s + 0.867·19-s + 0.914·21-s − 0.0469·22-s + 1.73·23-s − 0.609·24-s − 0.874·26-s + 0.192·27-s − 0.217·28-s + 0.509·29-s − 1.38·31-s + 0.272·32-s − 0.0292·33-s + 1.03·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(14.9719\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.592054772\)
\(L(\frac12)\) \(\approx\) \(3.592054772\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
good2 \( 1 - 1.31T + 2T^{2} \)
7 \( 1 - 4.19T + 7T^{2} \)
11 \( 1 + 0.167T + 11T^{2} \)
13 \( 1 + 3.39T + 13T^{2} \)
17 \( 1 - 4.57T + 17T^{2} \)
19 \( 1 - 3.78T + 19T^{2} \)
23 \( 1 - 8.31T + 23T^{2} \)
29 \( 1 - 2.74T + 29T^{2} \)
31 \( 1 + 7.71T + 31T^{2} \)
37 \( 1 - 2.07T + 37T^{2} \)
41 \( 1 + 1.28T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + 9.99T + 47T^{2} \)
53 \( 1 + 1.07T + 53T^{2} \)
59 \( 1 + 4.95T + 59T^{2} \)
61 \( 1 - 2.36T + 61T^{2} \)
67 \( 1 - 12.2T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 8.67T + 73T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + 4.06T + 89T^{2} \)
97 \( 1 + 2.78T + 97T^{2} \)
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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

−9.221738487804797920717557602432, −8.365083614769039401305153106934, −7.68059376986608340756711469214, −6.98007386845473149095295240974, −5.53487024852513171543505448509, −5.11705254067890107105669451845, −4.41851519283913319544985361436, −3.39939060259212624818149311914, −2.54841034615921783530298958530, −1.21537359839890350196407952427, 1.21537359839890350196407952427, 2.54841034615921783530298958530, 3.39939060259212624818149311914, 4.41851519283913319544985361436, 5.11705254067890107105669451845, 5.53487024852513171543505448509, 6.98007386845473149095295240974, 7.68059376986608340756711469214, 8.365083614769039401305153106934, 9.221738487804797920717557602432

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