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Properties

Label 1-1400-1400.1109-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.103 + 0.994i$
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  + (−0.913 − 0.406i)3-s + (0.669 + 0.743i)9-s + (−0.669 + 0.743i)11-s + (−0.309 + 0.951i)13-s + (−0.104 + 0.994i)17-s + (0.913 − 0.406i)19-s + (0.978 + 0.207i)23-s + (−0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (0.104 − 0.994i)31-s + (0.913 − 0.406i)33-s + (0.669 + 0.743i)37-s + (0.669 − 0.743i)39-s + (−0.309 + 0.951i)41-s + 43-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)3-s + (0.669 + 0.743i)9-s + (−0.669 + 0.743i)11-s + (−0.309 + 0.951i)13-s + (−0.104 + 0.994i)17-s + (0.913 − 0.406i)19-s + (0.978 + 0.207i)23-s + (−0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (0.104 − 0.994i)31-s + (0.913 − 0.406i)33-s + (0.669 + 0.743i)37-s + (0.669 − 0.743i)39-s + (−0.309 + 0.951i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.103 + 0.994i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.103 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9309538401 + 0.8391509749i\)
\(L(\frac12)\) \(\approx\) \(0.9309538401 + 0.8391509749i\)
\(L(1)\) \(\approx\) \(0.8094955551 + 0.07875177625i\)
\(L(1)\) \(\approx\) \(0.8094955551 + 0.07875177625i\)

Euler product

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

−20.61609435849302663752794216258, −19.74309296876458795114766993992, −18.69491413847439837757533454824, −18.07134464796360392702478377180, −17.49959641869493828781614721397, −16.50333454574271471418923157265, −15.99032886673426515418784369128, −15.38347993598789022519296990802, −14.35552308633346826712891059452, −13.51777191080749539769942055342, −12.56094879434390061863937065644, −12.0310325184068435083785663764, −10.93172030136073055253734115059, −10.63913590415551900615314494069, −9.66201068838420934259147702814, −8.88238771120709125256408423546, −7.74467167479605805510605605017, −7.0346475184620714744435521796, −5.97397426742756680304514639204, −5.27671923565340977465651164207, −4.72046632185739626669995617319, −3.41410188488702187163252961003, −2.754905670675586267119015135223, −1.08689260754134302891543682249, −0.3872142507143507848465189196, 0.8669405357955001493512419900, 1.8449142961077373053055878962, 2.78273044560970648911148277830, 4.26408502207800074488221853094, 4.826275920416969584878398109082, 5.77221169792445291644303655709, 6.595579321972389795404445363875, 7.340233074820018455373071495514, 8.04561679287468684099787757147, 9.27512210240289877947031035828, 10.04370255087941752163563425139, 10.83033431496195064952417209434, 11.66637831934098738583957566460, 12.211186724838342750286025683863, 13.16903236206027491741613319165, 13.60759070944386119964592877461, 14.88028630269165653342826217648, 15.48515825009612797568340752768, 16.43642243147053783061338564172, 17.0572770712670039262700417146, 17.73209207539229806036366836524, 18.46429877863246411774921063594, 19.1388985852675579739329333821, 19.91394926112662386924696688182, 20.931947254046504655982653624733

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