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Properties

Label 2-1300-65.29-c1-0-0
Degree $2$
Conductor $1300$
Sign $-0.848 - 0.529i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
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  + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (−1 + 1.73i)9-s + (−1.5 − 2.59i)11-s + (−3.46 + i)13-s + (−2.59 − 1.5i)17-s + (−3.5 + 6.06i)19-s − 0.999·21-s + (−2.59 + 1.5i)23-s + 5i·27-s + (1.5 + 2.59i)29-s − 4·31-s + (−2.59 − 1.5i)33-s + (6.06 − 3.5i)37-s + (−2.49 + 2.59i)39-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (−0.327 − 0.188i)7-s + (−0.333 + 0.577i)9-s + (−0.452 − 0.783i)11-s + (−0.960 + 0.277i)13-s + (−0.630 − 0.363i)17-s + (−0.802 + 1.39i)19-s − 0.218·21-s + (−0.541 + 0.312i)23-s + 0.962i·27-s + (0.278 + 0.482i)29-s − 0.718·31-s + (−0.452 − 0.261i)33-s + (0.996 − 0.575i)37-s + (−0.400 + 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.848 - 0.529i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ -0.848 - 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3027079486\)
\(L(\frac12)\) \(\approx\) \(0.3027079486\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (3.46 - i)T \)
good3 \( 1 + (-0.866 + 0.5i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-6.06 + 3.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.52 + 5.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.06 + 3.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.977430510011836634081639087598, −9.089082397325900261753823425972, −8.251020803882581368921321237200, −7.71785981066594500752696738125, −6.78768150669893101422966502028, −5.84293606616738709292195646779, −4.95285301094020354681251178557, −3.81711933791940795984021518520, −2.78655613054410225162071174764, −1.87258606616503860171160156213, 0.10555060916408512927287111008, 2.24071121244869294372185674253, 2.92200621383550730704894561856, 4.18803156068264208028469364552, 4.87985257266244360694493807909, 6.09571745741769660715039390942, 6.82932768186750260362354614040, 7.78618991333920241605941607284, 8.605345627917413113463336875319, 9.394554336568773483535723263778

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