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Properties

Label 2-1400-7.4-c1-0-26
Degree $2$
Conductor $1400$
Sign $0.376 + 0.926i$
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  + (1.02 − 1.77i)3-s + (2.18 + 1.48i)7-s + (−0.594 − 1.02i)9-s + (1.78 − 3.08i)11-s − 2.90·13-s + (1.85 − 3.22i)17-s + (1.78 + 3.09i)19-s + (4.87 − 2.35i)21-s + (−1.55 − 2.69i)23-s + 3.70·27-s + 1.57·29-s + (0.382 − 0.661i)31-s + (−3.64 − 6.31i)33-s + (−3.50 − 6.06i)37-s + (−2.97 + 5.15i)39-s + ⋯
L(s)  = 1  + (0.590 − 1.02i)3-s + (0.826 + 0.562i)7-s + (−0.198 − 0.343i)9-s + (0.537 − 0.930i)11-s − 0.806·13-s + (0.450 − 0.781i)17-s + (0.410 + 0.711i)19-s + (1.06 − 0.513i)21-s + (−0.324 − 0.561i)23-s + 0.713·27-s + 0.293·29-s + (0.0686 − 0.118i)31-s + (−0.634 − 1.09i)33-s + (−0.575 − 0.996i)37-s + (−0.476 + 0.825i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.376 + 0.926i$
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: \((2,\ 1400,\ (\ :1/2),\ 0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.325002529\)
\(L(\frac12)\) \(\approx\) \(2.325002529\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 1.48i)T \)
good3 \( 1 + (-1.02 + 1.77i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.78 + 3.08i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.90T + 13T^{2} \)
17 \( 1 + (-1.85 + 3.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.78 - 3.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.55 + 2.69i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.57T + 29T^{2} \)
31 \( 1 + (-0.382 + 0.661i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.50 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + (3.68 + 6.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.66 + 2.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.07 - 7.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.96 - 5.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.00 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + (-6.61 + 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.49 + 2.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (2.35 + 4.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−9.087183031595904462947065211380, −8.578364034936071995497347218263, −7.67525589944146292893320179044, −7.30998849688863579965237901187, −6.12968594732870738865773285933, −5.38372634416080179078489162833, −4.27244410936350985248639757611, −2.95351694682565724953628964762, −2.15227073444444670990384753191, −1.01149581969291093224360445107, 1.42130705257567573567985857661, 2.74814021753802372038502714019, 3.89331523584887167614216023205, 4.48984240116649023259729192012, 5.21001823230426543157691934826, 6.54315253696605512303091250943, 7.48338664555417750398864249538, 8.091637965543842586445903863772, 9.184359980670185194270271890473, 9.608526749683360560593820499125

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