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Properties

Label 2-1399-1399.1398-c0-0-11
Degree $2$
Conductor $1399$
Sign $-1$
Analytic cond. $0.698191$
Root an. cond. $0.835578$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.41i·3-s − 5-s + 1.41i·6-s + 8-s − 1.00·9-s + 10-s + 11-s − 1.41i·13-s + 1.41i·15-s − 16-s − 1.41i·17-s + 1.00·18-s − 19-s − 22-s + ⋯
L(s)  = 1  − 2-s − 1.41i·3-s − 5-s + 1.41i·6-s + 8-s − 1.00·9-s + 10-s + 11-s − 1.41i·13-s + 1.41i·15-s − 16-s − 1.41i·17-s + 1.00·18-s − 19-s − 22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1399\)
Sign: $-1$
Analytic conductor: \(0.698191\)
Root analytic conductor: \(0.835578\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1399} (1398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1399,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3586220254\)
\(L(\frac12)\) \(\approx\) \(0.3586220254\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1399 \( 1 + T \)
good2 \( 1 + T + T^{2} \)
3 \( 1 + 1.41iT - T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + 2T + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.975472364404858904423726395609, −8.492802323289460214512618669187, −7.76454743280948448637400801668, −7.19258120916628237640724830323, −6.58290407262445187501259559345, −5.25908781317726218332978192041, −4.16106191066540810119430273086, −2.93140411460478068329074948146, −1.51625136365408174606673498252, −0.45267078213554149236508111596, 1.74602925558783843878537537833, 3.69091674240504299262689881710, 4.18108279638917868502322751857, 4.67409704499359347329007205138, 6.19770267440204821360278490883, 7.08219734770414414027448340650, 8.260130387293976287694122039828, 8.598558974818364765186088558587, 9.433698976283179771884408078872, 9.910989459490278705825288386689

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