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Properties

Label 2-1400-35.9-c1-0-15
Degree $2$
Conductor $1400$
Sign $0.924 - 0.381i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.358 − 0.207i)3-s + (−2.09 − 1.62i)7-s + (−1.41 + 2.44i)9-s + (−0.414 − 0.717i)11-s + 2i·13-s + (6.63 − 3.82i)17-s + (−2.82 + 4.89i)19-s + (−1.08 − 0.148i)21-s + (4.83 + 2.79i)23-s + 2.41i·27-s + 7.82·29-s + (−0.414 − 0.717i)31-s + (−0.297 − 0.171i)33-s + (4.89 + 2.82i)37-s + (0.414 + 0.717i)39-s + ⋯
L(s)  = 1  + (0.207 − 0.119i)3-s + (−0.790 − 0.612i)7-s + (−0.471 + 0.816i)9-s + (−0.124 − 0.216i)11-s + 0.554i·13-s + (1.60 − 0.928i)17-s + (−0.648 + 1.12i)19-s + (−0.236 − 0.0324i)21-s + (1.00 + 0.582i)23-s + 0.464i·27-s + 1.45·29-s + (−0.0743 − 0.128i)31-s + (−0.0517 − 0.0298i)33-s + (0.805 + 0.464i)37-s + (0.0663 + 0.114i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.924 - 0.381i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.924 - 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.564866556\)
\(L(\frac12)\) \(\approx\) \(1.564866556\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.09 + 1.62i)T \)
good3 \( 1 + (-0.358 + 0.207i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.414 + 0.717i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (-6.63 + 3.82i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.82 - 4.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.83 - 2.79i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.82T + 29T^{2} \)
31 \( 1 + (0.414 + 0.717i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.89 - 2.82i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.82T + 41T^{2} \)
43 \( 1 + 6.89iT - 43T^{2} \)
47 \( 1 + (-10.0 - 5.82i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.89 - 2.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.32 - 5.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.1 + 6.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-3.16 + 1.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.75iT - 83T^{2} \)
89 \( 1 + (2.67 - 4.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−9.643460549396677899421614858085, −8.842400896287619845900583420544, −7.83208615699703979326234610689, −7.38098771567315304857327699747, −6.31832882408389615005374473412, −5.52449324332511089162904332679, −4.50514150338525675111339908422, −3.40438575837282232481371456678, −2.61534675489169185368230019054, −1.07091295373138915143719320760, 0.77667698561954778086707424390, 2.64420714062935690305849652945, 3.19678711673711918065649894028, 4.33600902521602101460603718778, 5.51096826311665397055340853838, 6.18225432035764762138680587310, 6.96143947699934824547318614111, 8.091664619601178868742232395666, 8.758985411678033994720462923150, 9.465333010182237803671656093422

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