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Properties

Label 2-1274-7.4-c1-0-3
Degree $2$
Conductor $1274$
Sign $-0.266 - 0.963i$
Analytic cond. $10.1729$
Root an. cond. $3.18950$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1 + 1.73i)5-s + 0.999·8-s + (1.5 + 2.59i)9-s + (0.999 − 1.73i)10-s + (−2 + 3.46i)11-s + 13-s + (−0.5 − 0.866i)16-s + (−3 + 5.19i)17-s + (1.5 − 2.59i)18-s − 1.99·20-s + 3.99·22-s + (−4 − 6.92i)23-s + (0.500 − 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.447 + 0.774i)5-s + 0.353·8-s + (0.5 + 0.866i)9-s + (0.316 − 0.547i)10-s + (−0.603 + 1.04i)11-s + 0.277·13-s + (−0.125 − 0.216i)16-s + (−0.727 + 1.26i)17-s + (0.353 − 0.612i)18-s − 0.447·20-s + 0.852·22-s + (−0.834 − 1.44i)23-s + (0.100 − 0.173i)25-s + (−0.0980 − 0.169i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (1145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9257463022\)
\(L(\frac12)\) \(\approx\) \(0.9257463022\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
13 \( 1 - T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.27218349280382840006829465646, −9.230312415844019715053843195373, −8.356034705758838103536782779553, −7.50008256785962660976668476360, −6.80379929589147692182202284651, −5.74189010230422693882198013864, −4.62064785055705089261799729370, −3.77700552042656540949213315800, −2.33090279808859789338708226288, −1.91724775583289348495911176473, 0.41836805644197940100525240865, 1.72209715873392458048317789276, 3.33654737723789333077148364178, 4.41815314042540122562323681403, 5.55152495605768014002451906284, 5.91068909008317905641105555610, 7.09358335571750465706097185668, 7.77184540525070484933981844702, 8.752067816849065704454512118625, 9.418081354915025642783831067549

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