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Properties

Label 2-1450-5.4-c1-0-24
Degree $2$
Conductor $1450$
Sign $0.894 + 0.447i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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 − 1.71i·3-s − 4-s + 1.71·6-s − 2.39i·7-s − i·8-s + 0.0460·9-s + 4.78·11-s + 1.71i·12-s + 4.50i·13-s + 2.39·14-s + 16-s + 0.391i·17-s + 0.0460i·18-s + 3.43·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.992i·3-s − 0.5·4-s + 0.701·6-s − 0.903i·7-s − 0.353i·8-s + 0.0153·9-s + 1.44·11-s + 0.496i·12-s + 1.24i·13-s + 0.639·14-s + 0.250·16-s + 0.0949i·17-s + 0.0108i·18-s + 0.788·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.812146499\)
\(L(\frac12)\) \(\approx\) \(1.812146499\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29 \( 1 + T \)
good3 \( 1 + 1.71iT - 3T^{2} \)
7 \( 1 + 2.39iT - 7T^{2} \)
11 \( 1 - 4.78T + 11T^{2} \)
13 \( 1 - 4.50iT - 13T^{2} \)
17 \( 1 - 0.391iT - 17T^{2} \)
19 \( 1 - 3.43T + 19T^{2} \)
23 \( 1 - 1.71iT - 23T^{2} \)
31 \( 1 + 3.73T + 31T^{2} \)
37 \( 1 - 2.78iT - 37T^{2} \)
41 \( 1 - 6.78T + 41T^{2} \)
43 \( 1 + 11.9iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 5.17iT - 53T^{2} \)
59 \( 1 - 9.93T + 59T^{2} \)
61 \( 1 - 3.71T + 61T^{2} \)
67 \( 1 + 6.87iT - 67T^{2} \)
71 \( 1 - 6.87T + 71T^{2} \)
73 \( 1 + 5.70iT - 73T^{2} \)
79 \( 1 + 9.19T + 79T^{2} \)
83 \( 1 - 4.56iT - 83T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 - 9.06iT - 97T^{2} \)
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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

−9.308380924457932943048188883407, −8.558719611546168003578187787141, −7.48258667355418637300747265737, −7.02632704228847592146167033831, −6.57598370080325968955703458909, −5.55631076152153701624938966774, −4.26892378561738989089006991207, −3.74131055120259071768490689669, −1.88532126883336692198103180128, −0.930832256054411639963939567755, 1.19489318023799982831494508720, 2.65157634116162382858051091589, 3.55563988735706322631387164454, 4.33085342391991569808303891285, 5.28793015902841182604453871371, 5.98543841824956079929424561640, 7.23283819124860804850605636347, 8.315424986017895422489078254884, 9.126527263506625059406342130776, 9.595347773669376706596528625763

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