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Properties

Label 2-1400-35.4-c1-0-25
Degree $2$
Conductor $1400$
Sign $-0.927 + 0.374i$
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  + (−2.09 − 1.20i)3-s + (0.358 − 2.62i)7-s + (1.41 + 2.44i)9-s + (2.41 − 4.18i)11-s − 2i·13-s + (−3.16 − 1.82i)17-s + (2.82 + 4.89i)19-s + (−3.91 + 5.04i)21-s + (7.28 − 4.20i)23-s + 0.414i·27-s + 2.17·29-s + (2.41 − 4.18i)31-s + (−10.0 + 5.82i)33-s + (−4.89 + 2.82i)37-s + (−2.41 + 4.18i)39-s + ⋯
L(s)  = 1  + (−1.20 − 0.696i)3-s + (0.135 − 0.990i)7-s + (0.471 + 0.816i)9-s + (0.727 − 1.26i)11-s − 0.554i·13-s + (−0.768 − 0.443i)17-s + (0.648 + 1.12i)19-s + (−0.854 + 1.10i)21-s + (1.51 − 0.877i)23-s + 0.0797i·27-s + 0.403·29-s + (0.433 − 0.751i)31-s + (−1.75 + 1.01i)33-s + (−0.805 + 0.464i)37-s + (−0.386 + 0.669i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.927 + 0.374i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9156319031\)
\(L(\frac12)\) \(\approx\) \(0.9156319031\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.358 + 2.62i)T \)
good3 \( 1 + (2.09 + 1.20i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.41 + 4.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (3.16 + 1.82i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.82 - 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.28 + 4.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.17T + 29T^{2} \)
31 \( 1 + (-2.41 + 4.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.89 - 2.82i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.171T + 41T^{2} \)
43 \( 1 + 12.8iT - 43T^{2} \)
47 \( 1 + (-0.297 + 0.171i)T + (23.5 - 40.7i)T^{2} \)
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 \( 1 + (-2.32 - 4.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.97 + 3.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (6.63 + 3.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.2iT - 83T^{2} \)
89 \( 1 + (8.32 + 14.4i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.132974910666384026400781657956, −8.332028872450189161422271132109, −7.29821710287988925842652193861, −6.76110398082440636900463214577, −5.96617951688216330495226004343, −5.22697653622864400879215416664, −4.15053429943336857045364467293, −3.07138665468091112339464242362, −1.29321925353138283183508222273, −0.50262941617084241181942546527, 1.53139078577151329687643499951, 2.90468708221799187657033659790, 4.38374028828674085778592929726, 4.84453033097021086525512889737, 5.63855126094579164530770917755, 6.61998269119917135313060879194, 7.14534837124433690748186864649, 8.584244502840768231389039124343, 9.307989125920612266279784955184, 9.809376559561690855267533425100

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