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Properties

Label 2-1300-65.2-c1-0-19
Degree $2$
Conductor $1300$
Sign $0.361 + 0.932i$
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  + (2.41 + 0.647i)3-s + (−3.34 − 1.92i)7-s + (2.82 + 1.63i)9-s + (1.17 − 4.38i)11-s + (−2.93 − 2.09i)13-s + (−1.34 − 5.00i)17-s + (6.75 − 1.80i)19-s + (−6.82 − 6.82i)21-s + (−0.490 + 1.82i)23-s + (0.465 + 0.465i)27-s + (−1.71 + 0.990i)29-s + (−2.74 + 2.74i)31-s + (5.67 − 9.83i)33-s + (10.2 − 5.93i)37-s + (−5.72 − 6.97i)39-s + ⋯
L(s)  = 1  + (1.39 + 0.373i)3-s + (−1.26 − 0.729i)7-s + (0.941 + 0.543i)9-s + (0.354 − 1.32i)11-s + (−0.812 − 0.582i)13-s + (−0.325 − 1.21i)17-s + (1.54 − 0.415i)19-s + (−1.48 − 1.48i)21-s + (−0.102 + 0.381i)23-s + (0.0895 + 0.0895i)27-s + (−0.318 + 0.183i)29-s + (−0.493 + 0.493i)31-s + (0.988 − 1.71i)33-s + (1.69 − 0.976i)37-s + (−0.916 − 1.11i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.361 + 0.932i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.361 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.057201921\)
\(L(\frac12)\) \(\approx\) \(2.057201921\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.93 + 2.09i)T \)
good3 \( 1 + (-2.41 - 0.647i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (3.34 + 1.92i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.17 + 4.38i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.34 + 5.00i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-6.75 + 1.80i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.490 - 1.82i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.71 - 0.990i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.74 - 2.74i)T - 31iT^{2} \)
37 \( 1 + (-10.2 + 5.93i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.04 - 1.88i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.07 - 1.35i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 2.88iT - 47T^{2} \)
53 \( 1 + (3.37 - 3.37i)T - 53iT^{2} \)
59 \( 1 + (1.22 + 4.58i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.59 + 2.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.60 - 2.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.70 - 6.36i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 3.35T + 73T^{2} \)
79 \( 1 + 0.191iT - 79T^{2} \)
83 \( 1 + 7.50iT - 83T^{2} \)
89 \( 1 + (-10.5 - 2.83i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.13 - 5.43i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.457415592647694386783565234036, −8.969395244789270911641562120799, −7.75694436133778622373811486211, −7.37081919445338128945729735311, −6.28028967873150379353057326499, −5.19134612991364774857721886222, −3.94870556059209226129832544480, −3.16740253327952246532051612434, −2.74465242212124558467383004647, −0.70178025254273693930564547446, 1.80752743042546095409127455800, 2.58266523999544128311542659668, 3.49786301424103595123957446496, 4.43954910520057956703176659305, 5.80145498488642678756822408503, 6.72495444104556048370569084724, 7.43801975563436485548157301524, 8.175944564942966406346030293666, 9.236366380705785581117347198845, 9.506943937802916675664973220496

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