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Properties

Label 2-1450-145.12-c1-0-14
Degree $2$
Conductor $1450$
Sign $0.0271 + 0.999i$
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 − 2.32·3-s − 4-s + 2.32i·6-s + (−2.17 − 2.17i)7-s + i·8-s + 2.40·9-s + (2.32 + 2.32i)11-s + 2.32·12-s + (0.506 + 0.506i)13-s + (−2.17 + 2.17i)14-s + 16-s + 0.0978i·17-s − 2.40i·18-s + (−2.42 + 2.42i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.34·3-s − 0.5·4-s + 0.949i·6-s + (−0.822 − 0.822i)7-s + 0.353i·8-s + 0.802·9-s + (0.701 + 0.701i)11-s + 0.671·12-s + (0.140 + 0.140i)13-s + (−0.581 + 0.581i)14-s + 0.250·16-s + 0.0237i·17-s − 0.567i·18-s + (−0.555 + 0.555i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0271 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0271 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $0.0271 + 0.999i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ 0.0271 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6939160789\)
\(L(\frac12)\) \(\approx\) \(0.6939160789\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 - 4.97i)T \)
good3 \( 1 + 2.32T + 3T^{2} \)
7 \( 1 + (2.17 + 2.17i)T + 7iT^{2} \)
11 \( 1 + (-2.32 - 2.32i)T + 11iT^{2} \)
13 \( 1 + (-0.506 - 0.506i)T + 13iT^{2} \)
17 \( 1 - 0.0978iT - 17T^{2} \)
19 \( 1 + (2.42 - 2.42i)T - 19iT^{2} \)
23 \( 1 + (1.50 - 1.50i)T - 23iT^{2} \)
31 \( 1 + (-0.164 - 0.164i)T + 31iT^{2} \)
37 \( 1 - 3.80T + 37T^{2} \)
41 \( 1 + (-6.66 + 6.66i)T - 41iT^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 - 5.14T + 47T^{2} \)
53 \( 1 + (-4.88 + 4.88i)T - 53iT^{2} \)
59 \( 1 - 0.622iT - 59T^{2} \)
61 \( 1 + (3.53 + 3.53i)T + 61iT^{2} \)
67 \( 1 + (2.53 - 2.53i)T - 67iT^{2} \)
71 \( 1 + 7.50iT - 71T^{2} \)
73 \( 1 + 9.39iT - 73T^{2} \)
79 \( 1 + (-9.18 + 9.18i)T - 79iT^{2} \)
83 \( 1 + (-5.71 + 5.71i)T - 83iT^{2} \)
89 \( 1 + (0.435 - 0.435i)T - 89iT^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−9.632244247383452801482452595817, −8.785501924937999733950695262634, −7.52888713218004722481500239473, −6.67182901159716674763773126638, −6.10261432389460241784785220678, −5.03662413871962371991676475321, −4.20398544652324078495402710638, −3.39453192531780065790201923738, −1.78324097691775277008388614006, −0.52861192798300442620235249410, 0.76969279810885630732773059615, 2.72878474374993223009684088080, 4.02997024233367369372690405737, 4.97849352012066416391726448258, 5.97443356355964987773963212723, 6.18088586600156182796583431299, 6.90213587527218030894653675951, 8.130445563258975526315947743181, 8.894443149995973310675522057498, 9.647779744845854077555545551299

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