Properties

Label 2-70e2-1.1-c1-0-8
Degree $2$
Conductor $4900$
Sign $1$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
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  − 1.17·3-s − 1.62·9-s + 6.62·11-s − 5.64·13-s − 7.99·17-s + 5.42·27-s − 0.623·29-s − 7.77·33-s + 6.62·39-s + 12.4·47-s + 9.37·51-s − 12·71-s + 13.4·73-s + 15.8·79-s − 1.49·81-s − 8.94·83-s + 0.731·87-s − 12.6·97-s − 10.7·99-s + 7.77·103-s − 9.87·109-s + 9.16·117-s + ⋯
L(s)  = 1  − 0.677·3-s − 0.541·9-s + 1.99·11-s − 1.56·13-s − 1.93·17-s + 1.04·27-s − 0.115·29-s − 1.35·33-s + 1.06·39-s + 1.81·47-s + 1.31·51-s − 1.42·71-s + 1.57·73-s + 1.78·79-s − 0.165·81-s − 0.981·83-s + 0.0784·87-s − 1.28·97-s − 1.08·99-s + 0.765·103-s − 0.945·109-s + 0.847·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(39.1266\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.073756859\)
\(L(\frac12)\) \(\approx\) \(1.073756859\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + 1.17T + 3T^{2} \)
11 \( 1 - 6.62T + 11T^{2} \)
13 \( 1 + 5.64T + 13T^{2} \)
17 \( 1 + 7.99T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 0.623T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−8.386180340092799930545843894106, −7.28390319063059513525944903807, −6.73491898727523643501237226985, −6.22179989661258799682394320593, −5.34648412163491855297776520707, −4.53490359577722223184028317113, −3.98856851767641320321434589048, −2.76352904329748997353639680725, −1.90613398862850428366013765502, −0.57441146660435581255081164576, 0.57441146660435581255081164576, 1.90613398862850428366013765502, 2.76352904329748997353639680725, 3.98856851767641320321434589048, 4.53490359577722223184028317113, 5.34648412163491855297776520707, 6.22179989661258799682394320593, 6.73491898727523643501237226985, 7.28390319063059513525944903807, 8.386180340092799930545843894106

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