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Properties

Label 2-1450-145.133-c1-0-13
Degree $2$
Conductor $1450$
Sign $-0.190 - 0.981i$
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 − 0.289·3-s − 4-s − 0.289i·6-s + (−0.632 + 0.632i)7-s − i·8-s − 2.91·9-s + (3.25 − 3.25i)11-s + 0.289·12-s + (−0.198 + 0.198i)13-s + (−0.632 − 0.632i)14-s + 16-s + 0.476i·17-s − 2.91i·18-s + (1.79 + 1.79i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.167·3-s − 0.5·4-s − 0.118i·6-s + (−0.239 + 0.239i)7-s − 0.353i·8-s − 0.972·9-s + (0.981 − 0.981i)11-s + 0.0836·12-s + (−0.0550 + 0.0550i)13-s + (−0.169 − 0.169i)14-s + 0.250·16-s + 0.115i·17-s − 0.687i·18-s + (0.411 + 0.411i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $-0.190 - 0.981i$
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: \((2,\ 1450,\ (\ :1/2),\ -0.190 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.241418644\)
\(L(\frac12)\) \(\approx\) \(1.241418644\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.30 - 0.918i)T \)
good3 \( 1 + 0.289T + 3T^{2} \)
7 \( 1 + (0.632 - 0.632i)T - 7iT^{2} \)
11 \( 1 + (-3.25 + 3.25i)T - 11iT^{2} \)
13 \( 1 + (0.198 - 0.198i)T - 13iT^{2} \)
17 \( 1 - 0.476iT - 17T^{2} \)
19 \( 1 + (-1.79 - 1.79i)T + 19iT^{2} \)
23 \( 1 + (-3.88 - 3.88i)T + 23iT^{2} \)
31 \( 1 + (5.65 - 5.65i)T - 31iT^{2} \)
37 \( 1 + 6.58T + 37T^{2} \)
41 \( 1 + (-6.47 - 6.47i)T + 41iT^{2} \)
43 \( 1 - 8.57T + 43T^{2} \)
47 \( 1 - 1.36T + 47T^{2} \)
53 \( 1 + (5.65 + 5.65i)T + 53iT^{2} \)
59 \( 1 - 9.89iT - 59T^{2} \)
61 \( 1 + (-3.88 + 3.88i)T - 61iT^{2} \)
67 \( 1 + (7.85 + 7.85i)T + 67iT^{2} \)
71 \( 1 - 8.63iT - 71T^{2} \)
73 \( 1 - 8.56iT - 73T^{2} \)
79 \( 1 + (0.0428 + 0.0428i)T + 79iT^{2} \)
83 \( 1 + (-1 - i)T + 83iT^{2} \)
89 \( 1 + (-5.88 - 5.88i)T + 89iT^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−9.338012949483512006265047898321, −8.963956497277987138207194794871, −8.201610734029262201878127255690, −7.24948152999173490378162203415, −6.35538876656130824280092124335, −5.78056040803242220881389004309, −5.00004641014641137723351253856, −3.72690929732037493997340295618, −2.95176867728438406720047130006, −1.11907273236633721528759515011, 0.61223786155932895025300196749, 2.08537980055689689577572416063, 3.09146571889776384456036910397, 4.11333302718312384419199319871, 4.94578662428127166656100325640, 5.94410749719364896871366507022, 6.87006994475756621952824513713, 7.68644762047870423408439626122, 8.934114812241924040281109605163, 9.171180056581513668464080786570

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