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Properties

Label 2-1300-65.4-c1-0-0
Degree $2$
Conductor $1300$
Sign $-0.375 - 0.927i$
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  + (−2.44 − 1.41i)3-s + (1.04 + 1.81i)7-s + (2.49 + 4.32i)9-s + (1.5 + 0.866i)11-s + (0.331 − 3.59i)13-s + (−3.14 + 1.81i)17-s + (−0.926 + 0.534i)19-s − 5.92i·21-s + (−6.77 − 3.90i)23-s − 5.62i·27-s + (−0.263 + 0.456i)29-s + 5.84i·31-s + (−2.44 − 4.24i)33-s + (4.87 − 8.44i)37-s + (−5.88 + 8.32i)39-s + ⋯
L(s)  = 1  + (−1.41 − 0.816i)3-s + (0.395 + 0.685i)7-s + (0.831 + 1.44i)9-s + (0.452 + 0.261i)11-s + (0.0918 − 0.995i)13-s + (−0.762 + 0.439i)17-s + (−0.212 + 0.122i)19-s − 1.29i·21-s + (−1.41 − 0.815i)23-s − 1.08i·27-s + (−0.0489 + 0.0847i)29-s + 1.04i·31-s + (−0.426 − 0.738i)33-s + (0.801 − 1.38i)37-s + (−0.942 + 1.33i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.375 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.375 - 0.927i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ -0.375 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3112247868\)
\(L(\frac12)\) \(\approx\) \(0.3112247868\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.331 + 3.59i)T \)
good3 \( 1 + (2.44 + 1.41i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.04 - 1.81i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.14 - 1.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.926 - 0.534i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.77 + 3.90i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.263 - 0.456i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.84iT - 31T^{2} \)
37 \( 1 + (-4.87 + 8.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.69 + 2.13i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.09 - 4.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 12.5iT - 53T^{2} \)
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 \( 1 + (5.41 - 9.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.2 - 7.08i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.64T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 + (-4.78 - 2.76i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.59 - 13.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.30220377584015357629353326185, −8.948905555598812708213463479152, −8.190269022923542262501582414143, −7.30679742572713924331046570400, −6.41614263418337112159685772804, −5.86574887798371225438472174369, −5.10732494758419265313296458804, −4.10219181807112605275174924556, −2.42766787173044470494694502104, −1.34407879008367188942025133066, 0.16402553179957364535387435571, 1.73351797843399501386692177036, 3.65486414735108444743906210927, 4.40982013457140575405507860654, 5.00101481333756784182006682864, 6.16088011973492253478638089288, 6.58439231318147390895129333703, 7.66478482601155005226258561640, 8.744579969046154861219271565519, 9.762360448316414369348535480078

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