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Properties

Label 2-950-95.74-c1-0-20
Degree $2$
Conductor $950$
Sign $0.999 - 0.0367i$
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  + (0.984 + 0.173i)2-s + (−0.223 + 0.266i)3-s + (0.939 + 0.342i)4-s + (−0.266 + 0.223i)6-s + (3.25 − 1.87i)7-s + (0.866 + 0.5i)8-s + (0.5 + 2.83i)9-s + (2.76 − 4.79i)11-s + (−0.300 + 0.173i)12-s + (0.839 + i)13-s + (3.53 − 1.28i)14-s + (0.766 + 0.642i)16-s + (−4.68 − 0.826i)17-s + 2.87i·18-s + (−2.77 − 3.35i)19-s + ⋯
L(s)  = 1  + (0.696 + 0.122i)2-s + (−0.128 + 0.153i)3-s + (0.469 + 0.171i)4-s + (−0.108 + 0.0911i)6-s + (1.23 − 0.710i)7-s + (0.306 + 0.176i)8-s + (0.166 + 0.945i)9-s + (0.833 − 1.44i)11-s + (−0.0868 + 0.0501i)12-s + (0.232 + 0.277i)13-s + (0.943 − 0.343i)14-s + (0.191 + 0.160i)16-s + (−1.13 − 0.200i)17-s + 0.678i·18-s + (−0.637 − 0.770i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.999 - 0.0367i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.999 - 0.0367i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74649 + 0.0505189i\)
\(L(\frac12)\) \(\approx\) \(2.74649 + 0.0505189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 - 0.173i)T \)
5 \( 1 \)
19 \( 1 + (2.77 + 3.35i)T \)
good3 \( 1 + (0.223 - 0.266i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-3.25 + 1.87i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.76 + 4.79i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.839 - i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (4.68 + 0.826i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.13 + 3.10i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.65 - 9.37i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.18 - 5.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + (-6.13 - 5.14i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.46 + 6.77i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-12.0 + 2.12i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.601 - 1.65i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.12 - 6.36i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-2.10 - 0.766i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (12.1 - 2.14i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.53 + 2.01i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (0.747 - 0.890i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (6.75 + 5.67i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (10.5 - 6.06i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.62 - 7.23i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (7.43 + 1.31i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.62162292805491684237123547219, −8.818630675506325193352534977158, −8.524942482361170560047978942476, −7.30829086955476375876632471964, −6.65853834331720682246047447121, −5.53696993997326333053735621457, −4.62682995710338738455341058775, −4.09201178644409796956972852881, −2.67382205488074314181219714030, −1.30939129116093651523726166871, 1.51187181015888876252346366405, 2.41727995158112454156951349218, 4.11335022813784043388061403128, 4.46918645277742577229969284841, 5.77044324634687163264722926418, 6.41011606396133059237673838033, 7.39269317950137560909036934003, 8.308867037725765394217862937480, 9.278122982470717944660649023117, 10.03111493205122649754435005513

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