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Properties

Label 2-950-95.12-c1-0-16
Degree $2$
Conductor $950$
Sign $0.646 + 0.762i$
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  + (−0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (0.499 − 0.866i)6-s + (−2.44 − 2.44i)7-s + (−0.707 − 0.707i)8-s + (1.73 + i)9-s + (−0.707 + 0.707i)12-s + (1.93 − 0.517i)13-s + (1.73 + 2.99i)14-s + (0.500 + 0.866i)16-s + (0.448 − 1.67i)17-s + (−1.41 − 1.41i)18-s + (−2.59 + 3.5i)19-s + (2.99 − 1.73i)21-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s + (0.204 − 0.353i)6-s + (−0.925 − 0.925i)7-s + (−0.249 − 0.249i)8-s + (0.577 + 0.333i)9-s + (−0.204 + 0.204i)12-s + (0.535 − 0.143i)13-s + (0.462 + 0.801i)14-s + (0.125 + 0.216i)16-s + (0.108 − 0.405i)17-s + (−0.333 − 0.333i)18-s + (−0.596 + 0.802i)19-s + (0.654 − 0.377i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.646 + 0.762i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.646 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.813817 - 0.376981i\)
\(L(\frac12)\) \(\approx\) \(0.813817 - 0.376981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
19 \( 1 + (2.59 - 3.5i)T \)
good3 \( 1 + (0.258 - 0.965i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.44 + 2.44i)T + 7iT^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-1.93 + 0.517i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.448 + 1.67i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (0.896 + 3.34i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.73 + 3i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + (-2.82 + 2.82i)T - 37iT^{2} \)
41 \( 1 + (-6 + 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.34 + 0.896i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-6.69 + 1.79i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.59 - 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.07 - 7.72i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-12 + 6.92i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.36 - 2.24i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.22 - 1.22i)T - 83iT^{2} \)
89 \( 1 + (-4.33 + 7.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.76 - 1.81i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.911895036436457774010215868764, −9.433975891198025567181668980132, −8.263640574446866352851801645956, −7.49623766517531248445002459399, −6.63338625728809193559299329095, −5.74389709174540342044075960256, −4.26012846986982287523106601677, −3.72696705082544811428387730906, −2.28521819819940436824847564018, −0.61791874041307553190663448562, 1.18288363846510395820940051922, 2.48483487106442689931828185303, 3.66716916458013185420928661777, 5.16025492946840469735120923890, 6.33620923767676564892996504618, 6.58056323673053613197695249869, 7.61751082674907249851292226978, 8.626364780099264172363961363229, 9.217987576984842553658773333834, 9.989118239193221536413477169635

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