Properties

Label 2-950-25.11-c1-0-40
Degree $2$
Conductor $950$
Sign $-0.997 - 0.0660i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (0.809 − 0.587i)2-s + (0.578 − 1.78i)3-s + (0.309 − 0.951i)4-s + (−1.13 − 1.92i)5-s + (−0.578 − 1.78i)6-s − 2.72·7-s + (−0.309 − 0.951i)8-s + (−0.409 − 0.297i)9-s + (−2.05 − 0.891i)10-s + (4.97 − 3.61i)11-s + (−1.51 − 1.10i)12-s + (−3.74 − 2.72i)13-s + (−2.20 + 1.59i)14-s + (−4.08 + 0.906i)15-s + (−0.809 − 0.587i)16-s + (1.58 + 4.88i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.334 − 1.02i)3-s + (0.154 − 0.475i)4-s + (−0.507 − 0.861i)5-s + (−0.236 − 0.727i)6-s − 1.02·7-s + (−0.109 − 0.336i)8-s + (−0.136 − 0.0992i)9-s + (−0.648 − 0.281i)10-s + (1.49 − 1.08i)11-s + (−0.437 − 0.317i)12-s + (−1.03 − 0.755i)13-s + (−0.588 + 0.427i)14-s + (−1.05 + 0.234i)15-s + (−0.202 − 0.146i)16-s + (0.384 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.997 - 0.0660i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.997 - 0.0660i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0592096 + 1.79076i\)
\(L(\frac12)\) \(\approx\) \(0.0592096 + 1.79076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (1.13 + 1.92i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.578 + 1.78i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 2.72T + 7T^{2} \)
11 \( 1 + (-4.97 + 3.61i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.74 + 2.72i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.58 - 4.88i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (2.94 - 2.14i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.142 - 0.437i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.363 - 1.11i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.180 - 0.131i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.45 + 1.05i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.58T + 43T^{2} \)
47 \( 1 + (-3.80 + 11.7i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.997 - 3.07i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.44 - 4.68i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-6.15 + 4.46i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.53 - 4.71i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.68 + 11.3i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.18 + 5.22i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.01 + 3.11i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.91 - 15.1i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-11.7 + 8.55i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.38 + 7.33i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.616840542808096126733817708765, −8.680821110512581513753815107742, −7.979793410351021405983652174781, −6.93721203853377830753696128729, −6.22519577101544365380891516823, −5.26187306460431560648326458577, −3.93086290191112225396837998084, −3.28414873560119372149066377930, −1.83240608224481032002686823209, −0.66654072770164721170003167143, 2.47574200903116624182144909613, 3.54004881498018791831238923286, 4.13456303841547227389223312399, 4.93081956671682761805388209754, 6.55583581720937163251409327575, 6.77706959383693763395021137949, 7.70583965669380019196676041818, 9.084710721645884328342819393496, 9.729969064738648744166470324664, 10.08684662715285767248673788815

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