Properties

Label 2-75-3.2-c14-0-42
Degree $2$
Conductor $75$
Sign $1$
Analytic cond. $93.2467$
Root an. cond. $9.65643$
Motivic weight $14$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.18e3·3-s + 1.63e4·4-s − 7.20e4·7-s + 4.78e6·9-s − 3.58e7·12-s + 3.33e7·13-s + 2.68e8·16-s + 6.90e7·19-s + 1.57e8·21-s − 1.04e10·27-s − 1.18e9·28-s + 5.47e10·31-s + 7.83e10·36-s − 1.11e11·37-s − 7.30e10·39-s − 3.75e11·43-s − 5.87e11·48-s − 6.73e11·49-s + 5.47e11·52-s − 1.51e11·57-s − 2.29e12·61-s − 3.44e11·63-s + 4.39e12·64-s + 4.34e12·67-s + 1.72e13·73-s + 1.13e12·76-s + 3.83e13·79-s + ⋯
L(s)  = 1  − 3-s + 4-s − 0.0875·7-s + 9-s − 12-s + 0.532·13-s + 16-s + 0.0772·19-s + 0.0875·21-s − 27-s − 0.0875·28-s + 1.99·31-s + 36-s − 1.17·37-s − 0.532·39-s − 1.38·43-s − 48-s − 0.992·49-s + 0.532·52-s − 0.0772·57-s − 0.729·61-s − 0.0875·63-s + 64-s + 0.717·67-s + 1.56·73-s + 0.0772·76-s + 1.99·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(93.2467\)
Root analytic conductor: \(9.65643\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: $\chi_{75} (26, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(2.112916330\)
\(L(\frac12)\) \(\approx\) \(2.112916330\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{7} T \)
5 \( 1 \)
good2 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
7 \( 1 + 72061 T + p^{14} T^{2} \)
11 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
13 \( 1 - 33388991 T + p^{14} T^{2} \)
17 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
19 \( 1 - 69090803 T + p^{14} T^{2} \)
23 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
29 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
31 \( 1 - 54792115547 T + p^{14} T^{2} \)
37 \( 1 + 111626070166 T + p^{14} T^{2} \)
41 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
43 \( 1 + 375951067189 T + p^{14} T^{2} \)
47 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
53 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
59 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
61 \( 1 + 2292094939633 T + p^{14} T^{2} \)
67 \( 1 - 4349633776379 T + p^{14} T^{2} \)
71 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
73 \( 1 - 17278443497906 T + p^{14} T^{2} \)
79 \( 1 - 38383122173618 T + p^{14} T^{2} \)
83 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
89 \( ( 1 - p^{7} T )( 1 + p^{7} T ) \)
97 \( 1 - 145408320661799 T + p^{14} T^{2} \)
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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

−11.65677291349244114275328181305, −10.77425299881233042622135199532, −9.876813087900949186991559124343, −8.128941090064110635039550683840, −6.85017876271259001993677684007, −6.15089571588694882684869893217, −4.92577683970823446766422743097, −3.39191858303923320636078736856, −1.87319506593651194901353894121, −0.76436014884102048101091786201, 0.76436014884102048101091786201, 1.87319506593651194901353894121, 3.39191858303923320636078736856, 4.92577683970823446766422743097, 6.15089571588694882684869893217, 6.85017876271259001993677684007, 8.128941090064110635039550683840, 9.876813087900949186991559124343, 10.77425299881233042622135199532, 11.65677291349244114275328181305

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