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Properties

Label 2-1300-65.29-c1-0-13
Degree $2$
Conductor $1300$
Sign $0.435 + 0.900i$
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  + (−2.59 + 1.5i)3-s + (2.59 + 1.5i)7-s + (3 − 5.19i)9-s + (−1.5 − 2.59i)11-s + (3.46 + i)13-s + (−6.06 − 3.5i)17-s + (0.5 − 0.866i)19-s − 9·21-s + (−6.06 + 3.5i)23-s + 9i·27-s + (−2.5 − 4.33i)29-s − 4·31-s + (7.79 + 4.5i)33-s + (2.59 − 1.5i)37-s + (−10.5 + 2.59i)39-s + ⋯
L(s)  = 1  + (−1.49 + 0.866i)3-s + (0.981 + 0.566i)7-s + (1 − 1.73i)9-s + (−0.452 − 0.783i)11-s + (0.960 + 0.277i)13-s + (−1.47 − 0.848i)17-s + (0.114 − 0.198i)19-s − 1.96·21-s + (−1.26 + 0.729i)23-s + 1.73i·27-s + (−0.464 − 0.804i)29-s − 0.718·31-s + (1.35 + 0.783i)33-s + (0.427 − 0.246i)37-s + (−1.68 + 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.435 + 0.900i$
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: \((2,\ 1300,\ (\ :1/2),\ 0.435 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5942707869\)
\(L(\frac12)\) \(\approx\) \(0.5942707869\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 1.5i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (6.06 + 3.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.06 - 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-2.59 + 1.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.79 - 4.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.2 - 6.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.52 + 5.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.561433960866115018130088914855, −8.868527764200207336072696153027, −7.957230359615789754279522098452, −6.76546306165468248768041626248, −5.86826651645782380958003358713, −5.40171219128121160443062457696, −4.53339359385173684778176438070, −3.72000397069170886026498023035, −2.03174172627228209767222686356, −0.32729642532591563108742734981, 1.23707753508114821971067660253, 2.09626907368375395436704493030, 4.13405823844536446109174462772, 4.77267191781205223333612541309, 5.80498441206477212824978277536, 6.38126906711376896659482517145, 7.31585496887357908714642008563, 7.88491431042209290981448554964, 8.808088525392697477932251842545, 10.24186344233696307244246316081

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