Properties

Label 2-700-1.1-c1-0-1
Degree $2$
Conductor $700$
Sign $1$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s − 2·9-s + 3·11-s + 13-s + 3·17-s + 2·19-s + 21-s + 6·23-s + 5·27-s − 9·29-s + 8·31-s − 3·33-s + 10·37-s − 39-s − 2·43-s + 3·47-s + 49-s − 3·51-s − 2·57-s + 12·59-s + 8·61-s + 2·63-s − 8·67-s − 6·69-s − 14·73-s − 3·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.277·13-s + 0.727·17-s + 0.458·19-s + 0.218·21-s + 1.25·23-s + 0.962·27-s − 1.67·29-s + 1.43·31-s − 0.522·33-s + 1.64·37-s − 0.160·39-s − 0.304·43-s + 0.437·47-s + 1/7·49-s − 0.420·51-s − 0.264·57-s + 1.56·59-s + 1.02·61-s + 0.251·63-s − 0.977·67-s − 0.722·69-s − 1.63·73-s − 0.341·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.202379852\)
\(L(\frac12)\) \(\approx\) \(1.202379852\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−10.52275298271203907560462125666, −9.551416552558793868042741249641, −8.871423639951469868858510987036, −7.80402753154155331215140932996, −6.76658613317771706160984607640, −5.98473030303912907687600422758, −5.17058967893037473212097767412, −3.90226018894727439504301993022, −2.82281852164300603542955300159, −0.992373800423296370916788271681, 0.992373800423296370916788271681, 2.82281852164300603542955300159, 3.90226018894727439504301993022, 5.17058967893037473212097767412, 5.98473030303912907687600422758, 6.76658613317771706160984607640, 7.80402753154155331215140932996, 8.871423639951469868858510987036, 9.551416552558793868042741249641, 10.52275298271203907560462125666

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