Properties

Label 2-2450-5.4-c1-0-42
Degree $2$
Conductor $2450$
Sign $0.894 - 0.447i$
Analytic cond. $19.5633$
Root an. cond. $4.42304$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 0.585i·3-s − 4-s − 0.585·6-s i·8-s + 2.65·9-s + 4.82·11-s − 0.585i·12-s + 0.828i·13-s + 16-s − 5.41i·17-s + 2.65i·18-s − 3.41·19-s + 4.82i·22-s − 6.82i·23-s + 0.585·24-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.338i·3-s − 0.5·4-s − 0.239·6-s − 0.353i·8-s + 0.885·9-s + 1.45·11-s − 0.169i·12-s + 0.229i·13-s + 0.250·16-s − 1.31i·17-s + 0.626i·18-s − 0.783·19-s + 1.02i·22-s − 1.42i·23-s + 0.119·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(19.5633\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2450} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.929117322\)
\(L(\frac12)\) \(\approx\) \(1.929117322\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - 0.585iT - 3T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 - 0.828iT - 13T^{2} \)
17 \( 1 + 5.41iT - 17T^{2} \)
19 \( 1 + 3.41T + 19T^{2} \)
23 \( 1 + 6.82iT - 23T^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 + 3.65iT - 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 3.17iT - 43T^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + 10.4iT - 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 - 9.65iT - 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 6.58iT - 73T^{2} \)
79 \( 1 - 1.17T + 79T^{2} \)
83 \( 1 - 6.24iT - 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 - 16.2iT - 97T^{2} \)
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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

−9.133188469852560898417838400406, −8.252169554957680857501380845436, −7.26084846068139564086523327622, −6.73825757635465499039326743097, −6.09143497799079737905451503493, −4.89893057642734328137393798047, −4.35300909641337723094536001973, −3.60473933700510773234109545827, −2.17954991242100634017009984744, −0.77166113878200625449024436728, 1.20516095868115670357061344905, 1.77503131842460740954774796660, 3.12584128703692414453406016990, 4.08159495995994266439823254636, 4.52926207254757590896453942987, 5.98611343077453752250912950188, 6.39399130804145172025830358645, 7.51843733102565442365396653653, 8.080914234065197501836451054057, 9.216775286300177306031598599612

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