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Properties

Label 2-1425-5.4-c1-0-30
Degree $2$
Conductor $1425$
Sign $0.894 - 0.447i$
Analytic cond. $11.3786$
Root an. cond. $3.37323$
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  + 2i·2-s + i·3-s − 2·4-s − 2·6-s − 3i·7-s − 9-s − 3·11-s − 2i·12-s − 6i·13-s + 6·14-s − 4·16-s − 3i·17-s − 2i·18-s + 19-s + 3·21-s − 6i·22-s + ⋯
L(s)  = 1  + 1.41i·2-s + 0.577i·3-s − 4-s − 0.816·6-s − 1.13i·7-s − 0.333·9-s − 0.904·11-s − 0.577i·12-s − 1.66i·13-s + 1.60·14-s − 16-s − 0.727i·17-s − 0.471i·18-s + 0.229·19-s + 0.654·21-s − 1.27i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(11.3786\)
Root analytic conductor: \(3.37323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1425} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1425,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.172221597\)
\(L(\frac12)\) \(\approx\) \(1.172221597\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
19 \( 1 - T \)
good2 \( 1 - 2iT - 2T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.553240512455051031707120729526, −8.413485375742797488947741712961, −7.895039796887645290250375378974, −7.25763668625528371800089456448, −6.41132631298482396721134952554, −5.24343815268340728410668430519, −5.08502283869155177745756381612, −3.79289298194400367263352114465, −2.74057995576155640585456380826, −0.48931966499224851519103196978, 1.34040613954526925788070271617, 2.34276376856678799762804501951, 2.88511316222343802985598495284, 4.21043114247154090773042122830, 5.08104433948292964566501808952, 6.30936337969802613601645509902, 6.88077912288907619054416324409, 8.321889040916067813821460774990, 8.684263139001281529221516510586, 9.693398029951034370896689740256

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