Properties

Label 1-1400-1400.347-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.481 + 0.876i$
Analytic cond. $6.50157$
Root an. cond. $6.50157$
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  + (−0.743 − 0.669i)3-s + (0.104 + 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.587 − 0.809i)13-s + (−0.207 − 0.978i)17-s + (−0.669 − 0.743i)19-s + (0.406 + 0.913i)23-s + (0.587 − 0.809i)27-s + (0.309 + 0.951i)29-s + (0.978 − 0.207i)31-s + (0.743 − 0.669i)33-s + (−0.994 + 0.104i)37-s + (−0.104 + 0.994i)39-s + (−0.809 + 0.587i)41-s i·43-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)3-s + (0.104 + 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.587 − 0.809i)13-s + (−0.207 − 0.978i)17-s + (−0.669 − 0.743i)19-s + (0.406 + 0.913i)23-s + (0.587 − 0.809i)27-s + (0.309 + 0.951i)29-s + (0.978 − 0.207i)31-s + (0.743 − 0.669i)33-s + (−0.994 + 0.104i)37-s + (−0.104 + 0.994i)39-s + (−0.809 + 0.587i)41-s i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.481 + 0.876i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.481 + 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1295661559 + 0.2189486256i\)
\(L(\frac12)\) \(\approx\) \(0.1295661559 + 0.2189486256i\)
\(L(1)\) \(\approx\) \(0.6605996620 - 0.08268214953i\)
\(L(1)\) \(\approx\) \(0.6605996620 - 0.08268214953i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.743 - 0.669i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (-0.207 - 0.978i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (0.406 + 0.913i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.994 + 0.104i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (0.743 + 0.669i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.994 + 0.104i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.96113132193117147066836711314, −19.69329936626741587335071640107, −19.022729568089729152775261137897, −18.30944202746960121786019683783, −17.06571741663124378994324738070, −17.00851121016575896476721713220, −16.09695422578154682507461535899, −15.30854922397974781329433698559, −14.58465175745622153162973974993, −13.761138500670796288237441953308, −12.70531688482021035090773964161, −11.99296206449926424277205515945, −11.2757434817238582901684832261, −10.440527775913225753559920797083, −9.95172985959266311491787716694, −8.7937769638115732302787898604, −8.28896441666129630120245295579, −6.866040840143555585990523955221, −6.268672064536703750306715008952, −5.47349408361444779072612081874, −4.4854778454825950144168336939, −3.878560275642815017879452713357, −2.7800640830448040494307594535, −1.51433533317230458125239468212, −0.11455700669741164402932078282, 1.189381984860893649136733741870, 2.23294612074527901821268457276, 3.09626427678719380492803454532, 4.709917204856160331512644900595, 4.99472393830592972520392359579, 6.078653746799032450036612343036, 7.06857256685198101137395063711, 7.393696478674472480865191950540, 8.45787066635429294521755884822, 9.51182401528753553252327422130, 10.35347158046927629375198388074, 11.063942759551797622847449388100, 12.01481754239855968623044991245, 12.47850923079482962850153098480, 13.3268536557245572020619418137, 13.9740229886010079897404797096, 15.22068473236514882419038385151, 15.60255806945261385332579078841, 16.77408094182056871420961202261, 17.39337999214781514217178878775, 17.90845850527401738694831681868, 18.64650226786898175956970517126, 19.590246810316468938143892850070, 20.08098219972441807251951408667, 21.0932967335875517535768157934

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