Properties

Label 2-7605-1.1-c1-0-82
Degree $2$
Conductor $7605$
Sign $1$
Analytic cond. $60.7262$
Root an. cond. $7.79270$
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  + 0.618·2-s − 1.61·4-s − 5-s + 4.23·7-s − 2.23·8-s − 0.618·10-s + 0.236·11-s + 2.61·14-s + 1.85·16-s + 3.47·17-s + 4.23·19-s + 1.61·20-s + 0.145·22-s − 3.76·23-s + 25-s − 6.85·28-s + 7.47·29-s + 5.61·32-s + 2.14·34-s − 4.23·35-s + 3·37-s + 2.61·38-s + 2.23·40-s − 11.9·41-s − 6.23·43-s − 0.381·44-s − 2.32·46-s + ⋯
L(s)  = 1  + 0.437·2-s − 0.809·4-s − 0.447·5-s + 1.60·7-s − 0.790·8-s − 0.195·10-s + 0.0711·11-s + 0.699·14-s + 0.463·16-s + 0.842·17-s + 0.971·19-s + 0.361·20-s + 0.0311·22-s − 0.784·23-s + 0.200·25-s − 1.29·28-s + 1.38·29-s + 0.993·32-s + 0.368·34-s − 0.716·35-s + 0.493·37-s + 0.424·38-s + 0.353·40-s − 1.86·41-s − 0.950·43-s − 0.0575·44-s − 0.342·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7605\)    =    \(3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(60.7262\)
Root analytic conductor: \(7.79270\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7605,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.340079912\)
\(L(\frac12)\) \(\approx\) \(2.340079912\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 0.618T + 2T^{2} \)
7 \( 1 - 4.23T + 7T^{2} \)
11 \( 1 - 0.236T + 11T^{2} \)
17 \( 1 - 3.47T + 17T^{2} \)
19 \( 1 - 4.23T + 19T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 - 7.47T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + 6.23T + 43T^{2} \)
47 \( 1 - 4.94T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 0.708T + 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 + 2.70T + 67T^{2} \)
71 \( 1 - 6.23T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + 3.47T + 97T^{2} \)
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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

−8.100412614685106831801772146647, −7.34146137357177518073743782989, −6.39254436004683582813404802107, −5.41962670096729503568232385547, −5.06872513047092423659189625960, −4.40764626336363909480021793246, −3.69868048639036736282884966731, −2.91226375364514096501580991219, −1.66547421798741475107881250981, −0.76781584062975058329137140400, 0.76781584062975058329137140400, 1.66547421798741475107881250981, 2.91226375364514096501580991219, 3.69868048639036736282884966731, 4.40764626336363909480021793246, 5.06872513047092423659189625960, 5.41962670096729503568232385547, 6.39254436004683582813404802107, 7.34146137357177518073743782989, 8.100412614685106831801772146647

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