Properties

Label 2-1300-13.10-c1-0-3
Degree $2$
Conductor $1300$
Sign $-0.895 + 0.444i$
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  + (1.56 + 2.70i)3-s + (−2.67 − 1.54i)7-s + (−3.39 + 5.88i)9-s + (−0.624 + 0.360i)11-s + (−1.64 − 3.20i)13-s + (−3.75 + 6.50i)17-s + (−6.43 − 3.71i)19-s − 9.67i·21-s + (2.83 + 4.91i)23-s − 11.8·27-s + (−1.20 − 2.08i)29-s − 4.05i·31-s + (−1.95 − 1.12i)33-s + (−0.624 + 0.360i)37-s + (6.11 − 9.47i)39-s + ⋯
L(s)  = 1  + (0.903 + 1.56i)3-s + (−1.01 − 0.584i)7-s + (−1.13 + 1.96i)9-s + (−0.188 + 0.108i)11-s + (−0.456 − 0.889i)13-s + (−0.910 + 1.57i)17-s + (−1.47 − 0.851i)19-s − 2.11i·21-s + (0.591 + 1.02i)23-s − 2.28·27-s + (−0.223 − 0.387i)29-s − 0.727i·31-s + (−0.340 − 0.196i)33-s + (−0.102 + 0.0592i)37-s + (0.979 − 1.51i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7310369881\)
\(L(\frac12)\) \(\approx\) \(0.7310369881\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 3.20i)T \)
good3 \( 1 + (-1.56 - 2.70i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.67 + 1.54i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.624 - 0.360i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.75 - 6.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.43 + 3.71i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.83 - 4.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.20 + 2.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.05iT - 31T^{2} \)
37 \( 1 + (0.624 - 0.360i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.11 - 1.80i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.55 - 4.42i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.51iT - 47T^{2} \)
53 \( 1 + 0.0561T + 53T^{2} \)
59 \( 1 + (1.28 + 0.739i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.37 - 5.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.8 + 6.24i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.77 - 5.06i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.14iT - 73T^{2} \)
79 \( 1 - 4.69T + 79T^{2} \)
83 \( 1 - 17.0iT - 83T^{2} \)
89 \( 1 + (-3.79 + 2.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.0 + 5.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.844261391431892805420060751307, −9.587939816032924769345856972926, −8.575038666035067426430917538457, −8.028425851187183148221401562571, −6.85114519016346816931148186743, −5.84825994949516896467398122145, −4.71778453453712357243135751270, −4.03153389581875214381411057959, −3.26737996070991672963963138809, −2.34186367213592854021074738498, 0.24155331631274153998768969944, 2.00395668814500299824677541372, 2.57728823935525054929128847421, 3.58210377555286021032266154095, 5.01238402595922612269503183165, 6.42856211909753828582017073776, 6.66132411182647408487828240080, 7.41671775920616634973275159718, 8.567272117425775329220039244112, 8.870135204884245271922983639648

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