Properties

Label 2-1400-1400.629-c0-0-0
Degree $2$
Conductor $1400$
Sign $-0.844 + 0.535i$
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  + (−0.951 − 0.309i)2-s + (−1.16 + 1.59i)3-s + (0.809 + 0.587i)4-s + (0.453 + 0.891i)5-s + (1.59 − 1.16i)6-s + i·7-s + (−0.587 − 0.809i)8-s + (−0.896 − 2.76i)9-s + (−0.156 − 0.987i)10-s + (−1.87 + 0.610i)12-s + (−0.297 + 0.0966i)13-s + (0.309 − 0.951i)14-s + (−1.95 − 0.309i)15-s + (0.309 + 0.951i)16-s + 2.90i·18-s + (−1.14 + 0.831i)19-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−1.16 + 1.59i)3-s + (0.809 + 0.587i)4-s + (0.453 + 0.891i)5-s + (1.59 − 1.16i)6-s + i·7-s + (−0.587 − 0.809i)8-s + (−0.896 − 2.76i)9-s + (−0.156 − 0.987i)10-s + (−1.87 + 0.610i)12-s + (−0.297 + 0.0966i)13-s + (0.309 − 0.951i)14-s + (−1.95 − 0.309i)15-s + (0.309 + 0.951i)16-s + 2.90i·18-s + (−1.14 + 0.831i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.844 + 0.535i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ -0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2888279911\)
\(L(\frac12)\) \(\approx\) \(0.2888279911\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-0.453 - 0.891i)T \)
7 \( 1 - iT \)
good3 \( 1 + (1.16 - 1.59i)T + (-0.309 - 0.951i)T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (1.14 - 0.831i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.550 + 1.69i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.280 - 0.863i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.533 + 0.734i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−10.12230641232092547735629902502, −9.743727817053203826658852297802, −8.989864871564318239008921833687, −8.130395880347899286488632344400, −6.65817683652675841058012926644, −6.16845833438832220551385804788, −5.45563783701951241810025982874, −4.19216960136911368666699894731, −3.28949455374398628859810863879, −2.17091458178505453995293788491, 0.36957256646041690495853063504, 1.45499583956826926060293213301, 2.30080141689865138661017673037, 4.60472924490394064749791934905, 5.47449460511366376868331451580, 6.29156176424379872643077555277, 6.80881436733102621876858989445, 7.72046013415697751984557947773, 8.135088661262604289624608088787, 9.159953744911655557939868526797

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