Properties

Label 2-950-475.111-c1-0-45
Degree $2$
Conductor $950$
Sign $-0.943 - 0.332i$
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  + (−0.997 − 0.0697i)2-s + (−0.0955 − 2.73i)3-s + (0.990 + 0.139i)4-s + (2.18 + 0.484i)5-s + (−0.0955 + 2.73i)6-s + (−1.98 − 3.44i)7-s + (−0.978 − 0.207i)8-s + (−4.49 + 0.314i)9-s + (−2.14 − 0.635i)10-s + (−0.154 − 1.47i)11-s + (0.286 − 2.72i)12-s + (−0.0682 − 0.101i)13-s + (1.74 + 3.57i)14-s + (1.11 − 6.02i)15-s + (0.961 + 0.275i)16-s + (−0.784 + 1.94i)17-s + ⋯
L(s)  = 1  + (−0.705 − 0.0493i)2-s + (−0.0551 − 1.58i)3-s + (0.495 + 0.0695i)4-s + (0.976 + 0.216i)5-s + (−0.0390 + 1.11i)6-s + (−0.751 − 1.30i)7-s + (−0.345 − 0.0735i)8-s + (−1.49 + 0.104i)9-s + (−0.677 − 0.200i)10-s + (−0.0466 − 0.443i)11-s + (0.0826 − 0.786i)12-s + (−0.0189 − 0.0280i)13-s + (0.465 + 0.954i)14-s + (0.288 − 1.55i)15-s + (0.240 + 0.0689i)16-s + (−0.190 + 0.471i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.943 - 0.332i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.943 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130772 + 0.764723i\)
\(L(\frac12)\) \(\approx\) \(0.130772 + 0.764723i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.997 + 0.0697i)T \)
5 \( 1 + (-2.18 - 0.484i)T \)
19 \( 1 + (2.33 - 3.67i)T \)
good3 \( 1 + (0.0955 + 2.73i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (1.98 + 3.44i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.154 + 1.47i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (0.0682 + 0.101i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (0.784 - 1.94i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (6.67 + 6.44i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (0.901 + 2.23i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (0.932 - 1.03i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (0.725 + 0.527i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.49 - 0.715i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (-1.17 + 6.65i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.40 - 5.94i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (-6.62 - 0.930i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (2.34 + 9.39i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (-2.82 - 2.72i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (3.97 + 2.48i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (7.15 + 3.80i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-8.16 + 12.1i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (-0.418 - 11.9i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (1.56 - 1.73i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (13.5 - 3.89i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (15.2 - 9.52i)T + (42.5 - 87.1i)T^{2} \)
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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

−9.670648862522164658254365410348, −8.541438600800135274769756511650, −7.84257073398241540002594922739, −6.98425232685046283286754327295, −6.37797192507271036936479397990, −5.86906087379718771526823668418, −3.93233191300132010576854061701, −2.56561103088750389617836774051, −1.63464261087162369448502151963, −0.43278144417303877699617820754, 2.13735492499269821979599922405, 3.05932476396629428998535615801, 4.39467540521028523729204227888, 5.50166682791579733790333129924, 5.90293814162624361883346823884, 7.08001580513138621331091601399, 8.588648495086708713261529470172, 9.063943295308881530835071451551, 9.755877010029636459234124520029, 9.998820036630813948661561825142

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