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Properties

Label 2-1300-13.10-c1-0-15
Degree $2$
Conductor $1300$
Sign $0.668 + 0.743i$
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  + (0.739 + 1.28i)3-s + (−2.71 − 1.56i)7-s + (0.406 − 0.704i)9-s + (−4.01 + 2.31i)11-s + (2.64 − 2.44i)13-s + (3.38 − 5.87i)17-s + (−1.45 − 0.839i)19-s − 4.63i·21-s + (2.76 + 4.79i)23-s + 5.63·27-s + (−3.87 − 6.70i)29-s − 1.46i·31-s + (−5.93 − 3.42i)33-s + (6.45 − 3.72i)37-s + (5.09 + 1.58i)39-s + ⋯
L(s)  = 1  + (0.426 + 0.739i)3-s + (−1.02 − 0.592i)7-s + (0.135 − 0.234i)9-s + (−1.20 + 0.698i)11-s + (0.734 − 0.678i)13-s + (0.821 − 1.42i)17-s + (−0.333 − 0.192i)19-s − 1.01i·21-s + (0.576 + 0.999i)23-s + 1.08·27-s + (−0.718 − 1.24i)29-s − 0.262i·31-s + (−1.03 − 0.596i)33-s + (1.06 − 0.612i)37-s + (0.815 + 0.253i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.668 + 0.743i$
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: \((2,\ 1300,\ (\ :1/2),\ 0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.441092729\)
\(L(\frac12)\) \(\approx\) \(1.441092729\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 2.44i)T \)
good3 \( 1 + (-0.739 - 1.28i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.71 + 1.56i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.01 - 2.31i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.38 + 5.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.45 + 0.839i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.76 - 4.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.87 + 6.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 + (-6.45 + 3.72i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.78 + 2.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.21 + 10.7i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.97iT - 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + (2.27 + 1.31i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.87 - 8.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.550 - 0.317i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.0 + 6.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.89iT - 73T^{2} \)
79 \( 1 - 6.21T + 79T^{2} \)
83 \( 1 - 3.33iT - 83T^{2} \)
89 \( 1 + (-2.27 + 1.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.43 + 3.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.503129033798341366464507010583, −9.118059930890420654852419861881, −7.67424862378978994759477299883, −7.39406916680703513176042090917, −6.14876222952519740592390271965, −5.27379386401518457279910998251, −4.22314037457443593757595928113, −3.40183925458906961185000406351, −2.61973075654203743074292736047, −0.59788711391475207618732431571, 1.38029103688406991017829754036, 2.62750435665376154761458009300, 3.34041826281785125293894316395, 4.65713128283724145226248459559, 5.94044298380819783790066914769, 6.30949612710422153937020931751, 7.41962338633861536680608872931, 8.207287039519695000546477344298, 8.725566747469174801057072944573, 9.690125926791154193020226270104

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