Properties

Label 2-3525-1.1-c1-0-102
Degree $2$
Conductor $3525$
Sign $-1$
Analytic cond. $28.1472$
Root an. cond. $5.30539$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.19·2-s + 3-s + 2.83·4-s − 2.19·6-s + 1.05·7-s − 1.82·8-s + 9-s − 3.22·11-s + 2.83·12-s + 2.58·13-s − 2.30·14-s − 1.64·16-s − 7.50·17-s − 2.19·18-s + 1.70·19-s + 1.05·21-s + 7.09·22-s − 2.21·23-s − 1.82·24-s − 5.69·26-s + 27-s + 2.97·28-s + 6.74·29-s + 10.8·31-s + 7.27·32-s − 3.22·33-s + 16.4·34-s + ⋯
L(s)  = 1  − 1.55·2-s + 0.577·3-s + 1.41·4-s − 0.897·6-s + 0.397·7-s − 0.645·8-s + 0.333·9-s − 0.972·11-s + 0.817·12-s + 0.718·13-s − 0.617·14-s − 0.412·16-s − 1.81·17-s − 0.518·18-s + 0.390·19-s + 0.229·21-s + 1.51·22-s − 0.461·23-s − 0.372·24-s − 1.11·26-s + 0.192·27-s + 0.562·28-s + 1.25·29-s + 1.94·31-s + 1.28·32-s − 0.561·33-s + 2.82·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-1$
Analytic conductor: \(28.1472\)
Root analytic conductor: \(5.30539\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
47 \( 1 + T \)
good2 \( 1 + 2.19T + 2T^{2} \)
7 \( 1 - 1.05T + 7T^{2} \)
11 \( 1 + 3.22T + 11T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
17 \( 1 + 7.50T + 17T^{2} \)
19 \( 1 - 1.70T + 19T^{2} \)
23 \( 1 + 2.21T + 23T^{2} \)
29 \( 1 - 6.74T + 29T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 + 5.81T + 41T^{2} \)
43 \( 1 + 3.39T + 43T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 + 6.55T + 61T^{2} \)
67 \( 1 - 0.750T + 67T^{2} \)
71 \( 1 - 2.09T + 71T^{2} \)
73 \( 1 - 4.80T + 73T^{2} \)
79 \( 1 - 6.59T + 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−8.402017995945642100693285102616, −7.899835073209759733045459578705, −6.83702434353551581696555149772, −6.50243762836923389176309017003, −5.08489768708014780695237765871, −4.36834708112709065118583040453, −3.07253277576104968253709304821, −2.22911314832556469154857253700, −1.37048080437691646416580379375, 0, 1.37048080437691646416580379375, 2.22911314832556469154857253700, 3.07253277576104968253709304821, 4.36834708112709065118583040453, 5.08489768708014780695237765871, 6.50243762836923389176309017003, 6.83702434353551581696555149772, 7.899835073209759733045459578705, 8.402017995945642100693285102616

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