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Properties

Label 2-1450-145.144-c1-0-24
Degree $2$
Conductor $1450$
Sign $-0.747 + 0.664i$
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  − 2-s + 4-s + 4i·7-s − 8-s − 3·9-s + 2i·11-s + 2i·13-s − 4i·14-s + 16-s − 6·17-s + 3·18-s − 2i·19-s − 2i·22-s − 2i·26-s + 4i·28-s + (−5 − 2i)29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.51i·7-s − 0.353·8-s − 9-s + 0.603i·11-s + 0.554i·13-s − 1.06i·14-s + 0.250·16-s − 1.45·17-s + 0.707·18-s − 0.458i·19-s − 0.426i·22-s − 0.392i·26-s + 0.755i·28-s + (−0.928 − 0.371i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $-0.747 + 0.664i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (1449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ -0.747 + 0.664i)\)

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 \( 1 + T \)
5 \( 1 \)
29 \( 1 + (5 + 2i)T \)
good3 \( 1 + 3T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 12iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 4iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.207126368420325773513589318187, −8.610554764270926428463810845226, −7.80802522795109585891807363799, −6.75748435688856633706368551465, −6.00598484504138557799423464722, −5.25733830887974062612019577414, −4.04546311717846069336166300152, −2.48112799891062184631540075983, −2.18931471190648924105222196916, 0, 1.26935074544523609893559452346, 2.78834305540247328660747891649, 3.68376905790042902442035312197, 4.80092857721606097335815709772, 5.94243492779152746842369568760, 6.71772745677472155631340712531, 7.51678849531419243953195012418, 8.312419754393057659997538172782, 8.892451888688794278217569519892

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