Properties

Label 1-1400-1400.667-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

Related objects

Downloads

Learn more

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: $\chi_{1400} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9353360662 - 1.580586725i\)
\(L(\frac12)\) \(\approx\) \(0.9353360662 - 1.580586725i\)
\(L(1)\) \(\approx\) \(1.184760673 - 0.5818244714i\)
\(L(1)\) \(\approx\) \(1.184760673 - 0.5818244714i\)

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 \( 1 + (0.951 + 0.309i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.11173359413164048783983955843, −20.239727667990666676048691945900, −19.721270191839644247922780338170, −18.902225070003218580249803714287, −18.11621162378871472657355919011, −17.12590868714272853329556120185, −16.428713061760264799823119912203, −15.65639342293320807092752324627, −14.87819640306962712531897918981, −14.4410301838696240202273476661, −13.42203513236890186442546291863, −12.80908461961949169081889496877, −11.79364921602291748580307681878, −10.79653995700490815861734455364, −10.19796012263571859153439296373, −9.38239685501027542877658696943, −8.60179931035465520597833334012, −7.9785161667227977544288189732, −6.89264793769496583794289356748, −6.108380025269962721719908626082, −4.68163445523567702329511654412, −4.41221784387101046861680882266, −3.34008586237253534267867566923, −2.36420517464463357674688492775, −1.545346101955172362965237804173, 0.627470855283692013600602582973, 1.62675809452773245848174902223, 2.80243023514007792233269092666, 3.35065956940343397919000972255, 4.385561190140724291611811986180, 5.772993351162691024236865717056, 6.215386328690419567927475582485, 7.40116625715063147743304170783, 8.04463771457677875491224650904, 8.66008626266449367417500317983, 9.57774950164553466349156181627, 10.43214021866370981262401880541, 11.44856296035377851453804020891, 12.16900052852251322136025319991, 13.04299191256003048420344431956, 13.793617399107378280921297298885, 14.131508683612584148406360790, 15.35695852096472894160361578648, 15.7374836852557207223619866954, 16.870582800114154689650000865497, 17.65316239885607825484010305942, 18.5386353058440203642687749861, 18.92302854763916265558230697715, 19.76031092328919172074935054542, 20.549989201597136950094722795885

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