Properties

Label 2-1400-35.13-c1-0-25
Degree $2$
Conductor $1400$
Sign $0.829 + 0.558i$
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  + (1.26 − 1.26i)3-s + (1.75 − 1.98i)7-s − 0.201i·9-s + 4.28·11-s + (−4.40 + 4.40i)13-s + (3.77 + 3.77i)17-s + 8.01·19-s + (−0.292 − 4.72i)21-s + (−2.07 − 2.07i)23-s + (3.54 + 3.54i)27-s − 0.383i·29-s − 1.01i·31-s + (5.41 − 5.41i)33-s + (5.30 − 5.30i)37-s + 11.1i·39-s + ⋯
L(s)  = 1  + (0.730 − 0.730i)3-s + (0.662 − 0.749i)7-s − 0.0670i·9-s + 1.29·11-s + (−1.22 + 1.22i)13-s + (0.914 + 0.914i)17-s + 1.83·19-s + (−0.0638 − 1.03i)21-s + (−0.432 − 0.432i)23-s + (0.681 + 0.681i)27-s − 0.0712i·29-s − 0.182i·31-s + (0.942 − 0.942i)33-s + (0.871 − 0.871i)37-s + 1.78i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.829 + 0.558i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.829 + 0.558i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.531841789\)
\(L(\frac12)\) \(\approx\) \(2.531841789\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 + 1.98i)T \)
good3 \( 1 + (-1.26 + 1.26i)T - 3iT^{2} \)
11 \( 1 - 4.28T + 11T^{2} \)
13 \( 1 + (4.40 - 4.40i)T - 13iT^{2} \)
17 \( 1 + (-3.77 - 3.77i)T + 17iT^{2} \)
19 \( 1 - 8.01T + 19T^{2} \)
23 \( 1 + (2.07 + 2.07i)T + 23iT^{2} \)
29 \( 1 + 0.383iT - 29T^{2} \)
31 \( 1 + 1.01iT - 31T^{2} \)
37 \( 1 + (-5.30 + 5.30i)T - 37iT^{2} \)
41 \( 1 - 3.42iT - 41T^{2} \)
43 \( 1 + (4.67 + 4.67i)T + 43iT^{2} \)
47 \( 1 + (6.51 + 6.51i)T + 47iT^{2} \)
53 \( 1 + (6.18 + 6.18i)T + 53iT^{2} \)
59 \( 1 + 9.39T + 59T^{2} \)
61 \( 1 - 1.99iT - 61T^{2} \)
67 \( 1 + (-0.224 + 0.224i)T - 67iT^{2} \)
71 \( 1 - 7.88T + 71T^{2} \)
73 \( 1 + (9.62 - 9.62i)T - 73iT^{2} \)
79 \( 1 - 8.59iT - 79T^{2} \)
83 \( 1 + (-6.32 + 6.32i)T - 83iT^{2} \)
89 \( 1 - 1.04T + 89T^{2} \)
97 \( 1 + (4.17 + 4.17i)T + 97iT^{2} \)
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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

−9.512949661020525866457714435635, −8.534374485436312535068652784086, −7.75964490086599242982557031055, −7.24009409914964273703051307766, −6.51244174713839470701936797675, −5.22111263601838749214794976798, −4.28244573888978581561070742919, −3.36729890710535830650166816313, −2.01414805803937381430256494864, −1.27336676937867465253430887839, 1.25598263000266371064861850290, 2.87401339236537766063284796182, 3.31479136648780936043898409716, 4.65048238587117296270586689019, 5.25145077683078923192261600322, 6.25794580970449140828535257671, 7.58712608541023313556867247177, 7.948128677991425167436186078273, 9.116888364054350115409199227543, 9.553242316753842275985678943694

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