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Properties

Label 2-1300-65.2-c1-0-0
Degree $2$
Conductor $1300$
Sign $-0.780 - 0.625i$
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  + (−0.834 − 0.223i)3-s + (−2.07 − 1.19i)7-s + (−1.95 − 1.12i)9-s + (0.0379 − 0.141i)11-s + (3.58 + 0.344i)13-s + (−1.33 − 4.96i)17-s + (4.18 − 1.12i)19-s + (1.46 + 1.46i)21-s + (−2.28 + 8.53i)23-s + (3.20 + 3.20i)27-s + (−5.00 + 2.88i)29-s + (−4.94 + 4.94i)31-s + (−0.0633 + 0.109i)33-s + (−3.76 + 2.17i)37-s + (−2.91 − 1.08i)39-s + ⋯
L(s)  = 1  + (−0.481 − 0.129i)3-s + (−0.783 − 0.452i)7-s + (−0.650 − 0.375i)9-s + (0.0114 − 0.0427i)11-s + (0.995 + 0.0956i)13-s + (−0.322 − 1.20i)17-s + (0.960 − 0.257i)19-s + (0.318 + 0.318i)21-s + (−0.476 + 1.77i)23-s + (0.617 + 0.617i)27-s + (−0.928 + 0.536i)29-s + (−0.887 + 0.887i)31-s + (−0.0110 + 0.0191i)33-s + (−0.618 + 0.356i)37-s + (−0.467 − 0.174i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.780 - 0.625i$
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: \((2,\ 1300,\ (\ :1/2),\ -0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1424793909\)
\(L(\frac12)\) \(\approx\) \(0.1424793909\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.58 - 0.344i)T \)
good3 \( 1 + (0.834 + 0.223i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (2.07 + 1.19i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0379 + 0.141i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.33 + 4.96i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.18 + 1.12i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.28 - 8.53i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (5.00 - 2.88i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.94 - 4.94i)T - 31iT^{2} \)
37 \( 1 + (3.76 - 2.17i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (11.6 + 3.11i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.642 - 0.172i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 3.88iT - 47T^{2} \)
53 \( 1 + (-2.38 + 2.38i)T - 53iT^{2} \)
59 \( 1 + (1.94 + 7.26i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.54 - 9.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.92 - 3.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.66 - 6.21i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 0.839T + 73T^{2} \)
79 \( 1 - 3.64iT - 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (11.7 + 3.15i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.254 + 0.440i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.847072010174902974935674966621, −9.255204853972802952954151834598, −8.466242066566888183736156004203, −7.20243130533489479011139048349, −6.82994997693334710850912102497, −5.71256425112217909233688698512, −5.19456948679265449960542572484, −3.65387585116336701046593365875, −3.16836898721357116799860473310, −1.39069946827864395313169225454, 0.06383695665394864105094917958, 1.93931164170902079681742283625, 3.17398777148999029596481443186, 4.08975525666589838602154107495, 5.31142724623866825894726428471, 6.04060853033373375560863305802, 6.53687581854829417818030496022, 7.85483255185733711483793639989, 8.552494879750109031260906183354, 9.290394122536700154209200615405

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