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Properties

Label 2-1450-5.4-c1-0-32
Degree $2$
Conductor $1450$
Sign $0.447 + 0.894i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 2i·3-s − 4-s − 2·6-s + 2i·7-s − i·8-s − 9-s − 6·11-s − 2i·12-s − 2i·13-s − 2·14-s + 16-s − 6i·17-s − i·18-s − 2·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.15i·3-s − 0.5·4-s − 0.816·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 1.80·11-s − 0.577i·12-s − 0.554i·13-s − 0.534·14-s + 0.250·16-s − 1.45i·17-s − 0.235i·18-s − 0.458·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $0.447 + 0.894i$
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: \(1\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 9T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 4iT - 97T^{2} \)
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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

−9.227711834250440085016337198682, −8.748367034588930178138098096136, −7.73932642641320336328375962718, −7.10600051731738631577252129864, −5.68196533244516170791628641172, −5.29642208402497731274704941599, −4.59916385054231776570974859258, −3.40470031884530021782360939367, −2.45875730195634571050949491854, 0, 1.47852189572173631258395816757, 2.27091093971533352857860902265, 3.47234528854097161946803218723, 4.50437246840661578070239812349, 5.51672628587367986842256871790, 6.56296677293455952721598660339, 7.30899608790696544734888275504, 8.115711287289803492281037559368, 8.601804173317114481841567989198

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