Properties

Label 2-1875-1.1-c3-0-178
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  − 4.95·2-s + 3·3-s + 16.5·4-s − 14.8·6-s + 28.2·7-s − 42.1·8-s + 9·9-s − 9.92·11-s + 49.5·12-s − 92.2·13-s − 140.·14-s + 76.7·16-s − 19.6·17-s − 44.5·18-s − 76.9·19-s + 84.8·21-s + 49.1·22-s + 101.·23-s − 126.·24-s + 456.·26-s + 27·27-s + 467.·28-s + 216.·29-s + 105.·31-s − 42.6·32-s − 29.7·33-s + 97.1·34-s + ⋯
L(s)  = 1  − 1.75·2-s + 0.577·3-s + 2.06·4-s − 1.01·6-s + 1.52·7-s − 1.86·8-s + 0.333·9-s − 0.271·11-s + 1.19·12-s − 1.96·13-s − 2.67·14-s + 1.19·16-s − 0.279·17-s − 0.583·18-s − 0.928·19-s + 0.881·21-s + 0.476·22-s + 0.918·23-s − 1.07·24-s + 3.44·26-s + 0.192·27-s + 3.15·28-s + 1.38·29-s + 0.608·31-s − 0.235·32-s − 0.156·33-s + 0.489·34-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 \( 1 + 4.95T + 8T^{2} \)
7 \( 1 - 28.2T + 343T^{2} \)
11 \( 1 + 9.92T + 1.33e3T^{2} \)
13 \( 1 + 92.2T + 2.19e3T^{2} \)
17 \( 1 + 19.6T + 4.91e3T^{2} \)
19 \( 1 + 76.9T + 6.85e3T^{2} \)
23 \( 1 - 101.T + 1.21e4T^{2} \)
29 \( 1 - 216.T + 2.43e4T^{2} \)
31 \( 1 - 105.T + 2.97e4T^{2} \)
37 \( 1 + 331.T + 5.06e4T^{2} \)
41 \( 1 - 95.1T + 6.89e4T^{2} \)
43 \( 1 + 67.8T + 7.95e4T^{2} \)
47 \( 1 - 117.T + 1.03e5T^{2} \)
53 \( 1 + 222.T + 1.48e5T^{2} \)
59 \( 1 - 174.T + 2.05e5T^{2} \)
61 \( 1 + 472.T + 2.26e5T^{2} \)
67 \( 1 + 125.T + 3.00e5T^{2} \)
71 \( 1 - 927.T + 3.57e5T^{2} \)
73 \( 1 - 101.T + 3.89e5T^{2} \)
79 \( 1 - 295.T + 4.93e5T^{2} \)
83 \( 1 + 1.11e3T + 5.71e5T^{2} \)
89 \( 1 - 127.T + 7.04e5T^{2} \)
97 \( 1 + 1.52e3T + 9.12e5T^{2} \)
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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

−8.427485808180079246715525097953, −7.986891660397888511322248162945, −7.26120635568868015463004223218, −6.66857828930753783382963548161, −5.12814220856127142891182375718, −4.49654808859184543460028944103, −2.73883394652600760940339775693, −2.16136858015930916408380123616, −1.23534025218651217113299305588, 0, 1.23534025218651217113299305588, 2.16136858015930916408380123616, 2.73883394652600760940339775693, 4.49654808859184543460028944103, 5.12814220856127142891182375718, 6.66857828930753783382963548161, 7.26120635568868015463004223218, 7.986891660397888511322248162945, 8.427485808180079246715525097953

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