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Properties

Label 2-1300-65.9-c1-0-14
Degree $2$
Conductor $1300$
Sign $-0.0854 + 0.996i$
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  + (−0.866 − 0.5i)3-s + (0.866 − 0.5i)7-s + (−1 − 1.73i)9-s + (−1.5 + 2.59i)11-s + (3.46 + i)13-s + (2.59 − 1.5i)17-s + (−3.5 − 6.06i)19-s − 0.999·21-s + (2.59 + 1.5i)23-s + 5i·27-s + (1.5 − 2.59i)29-s − 4·31-s + (2.59 − 1.5i)33-s + (−6.06 − 3.5i)37-s + (−2.49 − 2.59i)39-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.327 − 0.188i)7-s + (−0.333 − 0.577i)9-s + (−0.452 + 0.783i)11-s + (0.960 + 0.277i)13-s + (0.630 − 0.363i)17-s + (−0.802 − 1.39i)19-s − 0.218·21-s + (0.541 + 0.312i)23-s + 0.962i·27-s + (0.278 − 0.482i)29-s − 0.718·31-s + (0.452 − 0.261i)33-s + (−0.996 − 0.575i)37-s + (−0.400 − 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0854 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0854 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.0854 + 0.996i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ -0.0854 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.129721444\)
\(L(\frac12)\) \(\approx\) \(1.129721444\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.59 + 1.5i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (6.06 + 3.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.52 + 5.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.06 + 3.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.06 + 3.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.220411888873171838296621136479, −8.839879980444283491594291787306, −7.59983160904305868357105472997, −7.01292159022903193324567670057, −6.12149317775390873943957847124, −5.29258290183418749009594707175, −4.36167341590215850690254994013, −3.26383747401934582499251188272, −1.95404237055168856822359118301, −0.53978687998732208114433326842, 1.33959680066744382447069439720, 2.78971946111275467648817873811, 3.84004998829276107969003638542, 4.90976243768591211394641310509, 5.79306900546594112274890482786, 6.18484676600272839631296213837, 7.63945993262038558017452118044, 8.273156656691566090463133614658, 8.875187500964165111787003769419, 10.14483786014096906732266499352

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