Properties

Label 2-2450-5.4-c1-0-39
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 + 3.41i·3-s − 4-s − 3.41·6-s i·8-s − 8.65·9-s − 0.828·11-s − 3.41i·12-s − 4.82i·13-s + 16-s − 2.58i·17-s − 8.65i·18-s − 0.585·19-s − 0.828i·22-s − 1.17i·23-s + 3.41·24-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.97i·3-s − 0.5·4-s − 1.39·6-s − 0.353i·8-s − 2.88·9-s − 0.249·11-s − 0.985i·12-s − 1.33i·13-s + 0.250·16-s − 0.627i·17-s − 2.04i·18-s − 0.134·19-s − 0.176i·22-s − 0.244i·23-s + 0.696·24-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

\(L(1)\) \(\approx\) \(0.7360454048\)
\(L(\frac12)\) \(\approx\) \(0.7360454048\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41iT - 3T^{2} \)
11 \( 1 + 0.828T + 11T^{2} \)
13 \( 1 + 4.82iT - 13T^{2} \)
17 \( 1 + 2.58iT - 17T^{2} \)
19 \( 1 + 0.585T + 19T^{2} \)
23 \( 1 + 1.17iT - 23T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 - 7.65iT - 37T^{2} \)
41 \( 1 + 3.07T + 41T^{2} \)
43 \( 1 + 8.82iT - 43T^{2} \)
47 \( 1 + 5.17iT - 47T^{2} \)
53 \( 1 - 6.48iT - 53T^{2} \)
59 \( 1 + 8.58T + 59T^{2} \)
61 \( 1 - 9.31T + 61T^{2} \)
67 \( 1 + 1.65iT - 67T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 + 9.41iT - 73T^{2} \)
79 \( 1 - 6.82T + 79T^{2} \)
83 \( 1 + 2.24iT - 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 - 7.75iT - 97T^{2} \)
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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

−8.918237305750484599764712055217, −8.454196149736543836842264134802, −7.66890245398594656236150181842, −6.47256570128115876603700696546, −5.56335605917819474829950019010, −5.10599023679728606407900167558, −4.38935363964853185591959538580, −3.44368444177730901346324378500, −2.77465960323229942562283565960, −0.26729954105367526078283590755, 1.14694028055922218507332079025, 1.94693181847744733460579298353, 2.68033962629407582682350628222, 3.79276227776567919149712200088, 5.01702092665671347202660439597, 6.01108546143869933971750318286, 6.60890417629475776387088206919, 7.37944258220277072012878067610, 8.104972278800495397416264907702, 8.757102429514537163996759041468

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