Properties

Label 1-1400-1400.1021-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.535 - 0.844i$
Analytic cond. $150.450$
Root an. cond. $150.450$
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  + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)17-s + (0.309 − 0.951i)19-s + (−0.809 − 0.587i)23-s + (−0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.309 + 0.951i)33-s + (0.809 − 0.587i)37-s + (−0.809 − 0.587i)39-s + (0.809 − 0.587i)41-s − 43-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)17-s + (0.309 − 0.951i)19-s + (−0.809 − 0.587i)23-s + (−0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.309 + 0.951i)33-s + (0.809 − 0.587i)37-s + (−0.809 − 0.587i)39-s + (0.809 − 0.587i)41-s − 43-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7072446765 - 0.3888110511i\)
\(L(\frac12)\) \(\approx\) \(0.7072446765 - 0.3888110511i\)
\(L(1)\) \(\approx\) \(0.9334148384 + 0.3175090708i\)
\(L(1)\) \(\approx\) \(0.9334148384 + 0.3175090708i\)

Euler product

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

−20.33129455989551007745024758486, −20.0380900578038888378993278581, −19.24101112095034785137546295986, −18.40750525318206153198975620156, −17.93877727328330111378864068068, −16.95885107839370355210991432442, −16.39728102363954796116005560941, −15.21990385498524339144234755564, −14.47990033201911305248646042939, −13.87680585481142376557477339547, −13.133332038118474097562276972400, −12.22994781004546579316546459612, −11.73539920106649469133427778428, −10.854993442866764842754616355156, −9.582532131421398769187849570843, −9.14740320528942487524125174153, −7.85029454642562264865211900833, −7.67616377899430987427075895656, −6.47978732292892355204039193670, −5.90802251438857542550504889632, −4.85136569143875809892332775013, −3.59436454096263790707325150575, −2.87916679081009946613872027534, −1.82763512888390869194772039440, −0.92438803736292110675123494569, 0.15888953801234435134813576630, 1.809645332051935041781580852218, 2.58340098946627280761567126360, 3.79675563727339633867158346229, 4.3505339951317890962962103392, 5.15571163547101606545809265439, 6.237943755436415567850647708231, 7.09965320773258370348658499990, 8.12048016766411204772150248585, 8.97641109898715009617068295251, 9.59427309597984299569774770497, 10.282921206633332704760634652979, 11.20400184051364951299839267275, 11.90738289837323848289517781680, 12.81426969659445462763970271751, 13.86198029476122301883314877214, 14.5397934705889856359578451347, 15.09377198064896572018676719461, 15.88630921530337775689654603812, 16.73770282875340337045887495143, 17.2753287216848260019612716962, 18.09929309162350426413692488869, 19.36814973300891922285619517770, 19.759752989214126013336024241997, 20.37867456804599523123298395839

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