↑

Properties

Label 2-1450-145.133-c1-0-22
Degree $2$
Conductor $1450$
Sign $0.594 - 0.804i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 0.633·3-s − 4-s + 0.633i·6-s + (0.961 − 0.961i)7-s − i·8-s − 2.59·9-s + (−1.43 + 1.43i)11-s − 0.633·12-s + (3.31 − 3.31i)13-s + (0.961 + 0.961i)14-s + 16-s + 1.20i·17-s − 2.59i·18-s + (4.29 + 4.29i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.365·3-s − 0.5·4-s + 0.258i·6-s + (0.363 − 0.363i)7-s − 0.353i·8-s − 0.866·9-s + (−0.431 + 0.431i)11-s − 0.182·12-s + (0.920 − 0.920i)13-s + (0.256 + 0.256i)14-s + 0.250·16-s + 0.292i·17-s − 0.612i·18-s + (0.986 + 0.986i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $0.594 - 0.804i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (1293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ 0.594 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.812859820\)
\(L(\frac12)\) \(\approx\) \(1.812859820\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.90 - 4.53i)T \)
good3 \( 1 - 0.633T + 3T^{2} \)
7 \( 1 + (-0.961 + 0.961i)T - 7iT^{2} \)
11 \( 1 + (1.43 - 1.43i)T - 11iT^{2} \)
13 \( 1 + (-3.31 + 3.31i)T - 13iT^{2} \)
17 \( 1 - 1.20iT - 17T^{2} \)
19 \( 1 + (-4.29 - 4.29i)T + 19iT^{2} \)
23 \( 1 + (-0.821 - 0.821i)T + 23iT^{2} \)
31 \( 1 + (-6.99 + 6.99i)T - 31iT^{2} \)
37 \( 1 - 7.62T + 37T^{2} \)
41 \( 1 + (-1.02 - 1.02i)T + 41iT^{2} \)
43 \( 1 - 1.56T + 43T^{2} \)
47 \( 1 - 5.15T + 47T^{2} \)
53 \( 1 + (-6.99 - 6.99i)T + 53iT^{2} \)
59 \( 1 - 0.695iT - 59T^{2} \)
61 \( 1 + (-0.821 + 0.821i)T - 61iT^{2} \)
67 \( 1 + (9.48 + 9.48i)T + 67iT^{2} \)
71 \( 1 + 6.49iT - 71T^{2} \)
73 \( 1 + 7.91iT - 73T^{2} \)
79 \( 1 + (-8.50 - 8.50i)T + 79iT^{2} \)
83 \( 1 + (-1 - i)T + 83iT^{2} \)
89 \( 1 + (9.73 + 9.73i)T + 89iT^{2} \)
97 \( 1 + 2.86T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.482484738221290633598566631623, −8.635226654878628554097348101781, −7.85120503153709623995277963630, −7.60298078398744623361808442181, −6.18519581179558348585907977000, −5.71427459413128169963905484296, −4.69988100609565775436924567089, −3.66394881429150516561112340260, −2.71202288011374665826810132334, −1.03999093383918903166549917810, 0.936684669241876736118130492489, 2.41690665338965338281321164007, 3.03584414989405475988866046404, 4.17101317344356920953743185274, 5.13989035572464864365079241319, 5.96057947140903716343737399104, 7.01891214487603693941589932042, 8.206478988903187000982420776857, 8.641498469684408373152398904099, 9.313162947326486798407278850556

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