Properties

Label 2-1400-7.4-c1-0-2
Degree $2$
Conductor $1400$
Sign $-0.991 + 0.126i$
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  + (−1 + 1.73i)3-s + (−2 + 1.73i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + 3·13-s + (−1 + 1.73i)17-s + (2.5 + 4.33i)19-s + (−0.999 − 5.19i)21-s + (3.5 + 6.06i)23-s − 4.00·27-s − 6·29-s + (−2 + 3.46i)31-s + (0.999 + 1.73i)33-s + (−2.5 − 4.33i)37-s + (−3 + 5.19i)39-s + ⋯
L(s)  = 1  + (−0.577 + 0.999i)3-s + (−0.755 + 0.654i)7-s + (−0.166 − 0.288i)9-s + (0.150 − 0.261i)11-s + 0.832·13-s + (−0.242 + 0.420i)17-s + (0.573 + 0.993i)19-s + (−0.218 − 1.13i)21-s + (0.729 + 1.26i)23-s − 0.769·27-s − 1.11·29-s + (−0.359 + 0.622i)31-s + (0.174 + 0.301i)33-s + (−0.410 − 0.711i)37-s + (−0.480 + 0.832i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7368331701\)
\(L(\frac12)\) \(\approx\) \(0.7368331701\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good3 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (-6 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−10.08921634039807693777943196528, −9.218232201703958353331722765946, −8.686558791093811603843156092851, −7.53348643410263489050637756703, −6.51431378284343142968756851234, −5.59701052952267362423131515776, −5.24381242950577318557947562524, −3.78701812917989871350363925335, −3.42593979867028206654495862237, −1.72741785130060712862589600306, 0.33464136762397425420927785505, 1.43443711147013029053496661736, 2.87048462374737838085946338544, 3.94112991422782330077792513114, 5.04317126478487142612273099994, 6.11255172395839609847995573404, 6.80617617349166798761665443841, 7.16885078576524821138669290795, 8.213883586280911004510475687181, 9.227195249067869121867676290485

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