Properties

Label 2-1400-280.69-c0-0-4
Degree $2$
Conductor $1400$
Sign $-0.447 + 0.894i$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
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  i·2-s − 4-s i·7-s + i·8-s + 9-s − 14-s + 16-s i·18-s − 2i·23-s + i·28-s i·32-s − 36-s − 2·46-s − 49-s + 56-s + ⋯
L(s)  = 1  i·2-s − 4-s i·7-s + i·8-s + 9-s − 14-s + 16-s i·18-s − 2i·23-s + i·28-s i·32-s − 36-s − 2·46-s − 49-s + 56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.012088152\)
\(L(\frac12)\) \(\approx\) \(1.012088152\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 2iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
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

−9.765012748437688413368988412335, −8.866905105064671329983701953689, −8.002590247929937488962729954525, −7.16429588126950139333433034383, −6.23411258645905093824127678128, −4.83873280622959609951008350304, −4.32135749502739859126499642656, −3.42991126237455350429414279205, −2.19301970758838871135275432597, −0.954165853816513525998763122907, 1.62674609263186377695371728245, 3.24988279972240346684076149846, 4.27687517674649775287371903480, 5.22797957112767086184052552108, 5.87814879234559160367791851818, 6.81684790072022592688546797161, 7.55305362595740159195052512078, 8.278973365732278437695112623326, 9.273086296306183427017120347383, 9.602136215530039227366546270583

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