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Properties

Label 2-1300-13.10-c1-0-11
Degree $2$
Conductor $1300$
Sign $-0.0791 - 0.996i$
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  + (1.41 + 2.44i)3-s + (1.81 + 1.04i)7-s + (−2.49 + 4.32i)9-s + (1.5 − 0.866i)11-s + (3.59 − 0.331i)13-s + (1.81 − 3.14i)17-s + (0.926 + 0.534i)19-s + 5.92i·21-s + (3.90 + 6.77i)23-s − 5.62·27-s + (0.263 + 0.456i)29-s − 5.84i·31-s + (4.24 + 2.44i)33-s + (−8.44 + 4.87i)37-s + (5.88 + 8.32i)39-s + ⋯
L(s)  = 1  + (0.816 + 1.41i)3-s + (0.685 + 0.395i)7-s + (−0.831 + 1.44i)9-s + (0.452 − 0.261i)11-s + (0.995 − 0.0918i)13-s + (0.439 − 0.762i)17-s + (0.212 + 0.122i)19-s + 1.29i·21-s + (0.815 + 1.41i)23-s − 1.08·27-s + (0.0489 + 0.0847i)29-s − 1.04i·31-s + (0.738 + 0.426i)33-s + (−1.38 + 0.801i)37-s + (0.942 + 1.33i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.0791 - 0.996i$
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: \((2,\ 1300,\ (\ :1/2),\ -0.0791 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.530271915\)
\(L(\frac12)\) \(\approx\) \(2.530271915\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 + 0.331i)T \)
good3 \( 1 + (-1.41 - 2.44i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.81 - 1.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.81 + 3.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.926 - 0.534i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.90 - 6.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.263 - 0.456i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.84iT - 31T^{2} \)
37 \( 1 + (8.44 - 4.87i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.69 - 2.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.67 - 8.09i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + (1.21 + 0.701i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.55 + 9.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.38 + 5.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.2 + 7.08i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.64iT - 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + 15.7iT - 83T^{2} \)
89 \( 1 + (4.78 - 2.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.1 - 7.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.619557930705096063276900335749, −9.218504429947896448690505571536, −8.371647218240285722734297939702, −7.82161368708201884769249884485, −6.49152148289764570428868587687, −5.30756694592498334338400895862, −4.80228336511858842578402483564, −3.58608494490015767920456394889, −3.16207095982930803001104818946, −1.62668990578418939230222804524, 1.11684143707481314357475385776, 1.84564271996858583605448814497, 3.08689495825043471410033749603, 4.06103505812584032510054882119, 5.32382278131493428586176481831, 6.54672813337733903886661934985, 6.94768332050399799914564228088, 7.88180721074273551709123697845, 8.539365602241681585453061221485, 8.985005732669105115833726395340

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