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Properties

Label 2-950-19.7-c1-0-28
Degree $2$
Conductor $950$
Sign $0.634 + 0.772i$
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  + (0.5 + 0.866i)2-s + (−0.341 − 0.590i)3-s + (−0.499 + 0.866i)4-s + (0.341 − 0.590i)6-s + 0.317·7-s − 0.999·8-s + (1.26 − 2.19i)9-s − 4.31·11-s + 0.682·12-s + (3.14 − 5.45i)13-s + (0.158 + 0.275i)14-s + (−0.5 − 0.866i)16-s + (−0.0669 − 0.115i)17-s + 2.53·18-s + (−4.05 + 1.60i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.196 − 0.341i)3-s + (−0.249 + 0.433i)4-s + (0.139 − 0.241i)6-s + 0.120·7-s − 0.353·8-s + (0.422 − 0.731i)9-s − 1.29·11-s + 0.196·12-s + (0.873 − 1.51i)13-s + (0.0424 + 0.0735i)14-s + (−0.125 − 0.216i)16-s + (−0.0162 − 0.0281i)17-s + 0.597·18-s + (−0.929 + 0.367i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.634 + 0.772i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.634 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28626 - 0.608098i\)
\(L(\frac12)\) \(\approx\) \(1.28626 - 0.608098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
19 \( 1 + (4.05 - 1.60i)T \)
good3 \( 1 + (0.341 + 0.590i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 0.317T + 7T^{2} \)
11 \( 1 + 4.31T + 11T^{2} \)
13 \( 1 + (-3.14 + 5.45i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.0669 + 0.115i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.98 + 3.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.57 + 7.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.98T + 31T^{2} \)
37 \( 1 - 5.07T + 37T^{2} \)
41 \( 1 + (-0.433 - 0.750i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.85 - 4.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.48 + 11.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.96 + 6.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.80 - 8.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.08 - 5.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.295 - 0.511i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.83 + 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.13 + 7.15i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.66 - 2.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.20T + 83T^{2} \)
89 \( 1 + (-1.85 + 3.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.42 - 4.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.08572522589472775816625155463, −8.788761849173081201800579188646, −8.078834696170898540136754426437, −7.43737069621388785023075889329, −6.30047723561498522982709702207, −5.81451358895641842831881050268, −4.75334676379932561935842561954, −3.68936802865848144996659460563, −2.54543417239482242348540013503, −0.61929477222306742565441143363, 1.59818997398290813721804145413, 2.69226422386638685455650060607, 4.02022511677176568946484857450, 4.73783885801683401808825832046, 5.56005296878172454271601986019, 6.66393602743403591089810825464, 7.65217710285542465102181895617, 8.699315258484583372532414819968, 9.423824733533889413577614054926, 10.50326853769348462667776354108

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