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Properties

Label 2-1400-35.27-c1-0-2
Degree $2$
Conductor $1400$
Sign $0.502 - 0.864i$
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 − 2.09i)3-s + (0.510 − 2.59i)7-s + 5.75i·9-s − 3.94·11-s + (1.69 + 1.69i)13-s + (2.66 − 2.66i)17-s − 5.36·19-s + (−6.50 + 4.36i)21-s + (−3.55 + 3.55i)23-s + (5.77 − 5.77i)27-s + 7.24i·29-s − 0.174i·31-s + (8.26 + 8.26i)33-s + (−4.85 − 4.85i)37-s − 7.10i·39-s + ⋯
L(s)  = 1  + (−1.20 − 1.20i)3-s + (0.193 − 0.981i)7-s + 1.91i·9-s − 1.19·11-s + (0.470 + 0.470i)13-s + (0.647 − 0.647i)17-s − 1.22·19-s + (−1.41 + 0.952i)21-s + (−0.741 + 0.741i)23-s + (1.11 − 1.11i)27-s + 1.34i·29-s − 0.0312i·31-s + (1.43 + 1.43i)33-s + (−0.798 − 0.798i)37-s − 1.13i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.502 - 0.864i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3018708235\)
\(L(\frac12)\) \(\approx\) \(0.3018708235\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.510 + 2.59i)T \)
good3 \( 1 + (2.09 + 2.09i)T + 3iT^{2} \)
11 \( 1 + 3.94T + 11T^{2} \)
13 \( 1 + (-1.69 - 1.69i)T + 13iT^{2} \)
17 \( 1 + (-2.66 + 2.66i)T - 17iT^{2} \)
19 \( 1 + 5.36T + 19T^{2} \)
23 \( 1 + (3.55 - 3.55i)T - 23iT^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + 0.174iT - 31T^{2} \)
37 \( 1 + (4.85 + 4.85i)T + 37iT^{2} \)
41 \( 1 + 0.732iT - 41T^{2} \)
43 \( 1 + (8.20 - 8.20i)T - 43iT^{2} \)
47 \( 1 + (-2.31 + 2.31i)T - 47iT^{2} \)
53 \( 1 + (-4.53 + 4.53i)T - 53iT^{2} \)
59 \( 1 - 13.1T + 59T^{2} \)
61 \( 1 - 13.3iT - 61T^{2} \)
67 \( 1 + (-6.55 - 6.55i)T + 67iT^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 + (7.02 + 7.02i)T + 73iT^{2} \)
79 \( 1 - 6.63iT - 79T^{2} \)
83 \( 1 + (10.4 + 10.4i)T + 83iT^{2} \)
89 \( 1 + 3.43T + 89T^{2} \)
97 \( 1 + (1.40 - 1.40i)T - 97iT^{2} \)
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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

−10.08826660095922466102408870146, −8.623570893388848733816361027379, −7.80239091997372615129651501333, −7.16307597916174559945082459483, −6.57372619190383183180456311658, −5.57191914817703951792687207072, −4.97328856479332024412343971386, −3.73977950477889078161590768077, −2.18244687029566221668510323637, −1.10676619645497212223845527801, 0.16096724820307897723200166723, 2.24704889718640919614437191685, 3.55821488394553775964856846117, 4.50578668389137308654634517184, 5.36066718630498996064086466658, 5.82556307876303394229259098354, 6.60060039566177619467874123450, 8.184639139070146723810228337804, 8.515023135295797362862683853190, 9.834290595331177987162856433880

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