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Properties

Label 2-950-25.11-c1-0-43
Degree $2$
Conductor $950$
Sign $-0.876 - 0.481i$
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 + (1 − 3.07i)3-s + (0.309 − 0.951i)4-s + (0.690 + 2.12i)5-s + (−1 − 3.07i)6-s − 5.23·7-s + (−0.309 − 0.951i)8-s + (−6.04 − 4.39i)9-s + (1.80 + 1.31i)10-s + (−1.61 + 1.17i)11-s + (−2.61 − 1.90i)12-s + (−1.5 − 1.08i)13-s + (−4.23 + 3.07i)14-s + 7.23·15-s + (−0.809 − 0.587i)16-s + (−0.5 − 1.53i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.577 − 1.77i)3-s + (0.154 − 0.475i)4-s + (0.309 + 0.951i)5-s + (−0.408 − 1.25i)6-s − 1.97·7-s + (−0.109 − 0.336i)8-s + (−2.01 − 1.46i)9-s + (0.572 + 0.415i)10-s + (−0.487 + 0.354i)11-s + (−0.755 − 0.549i)12-s + (−0.416 − 0.302i)13-s + (−1.13 + 0.822i)14-s + 1.86·15-s + (−0.202 − 0.146i)16-s + (−0.121 − 0.373i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.876 - 0.481i$
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: \((2,\ 950,\ (\ :1/2),\ -0.876 - 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.326465 + 1.27149i\)
\(L(\frac12)\) \(\approx\) \(0.326465 + 1.27149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.690 - 2.12i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-1 + 3.07i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 5.23T + 7T^{2} \)
11 \( 1 + (1.61 - 1.17i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.5 + 1.08i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (-6.85 + 4.97i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.263 - 0.812i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.61 + 4.97i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.54 + 3.30i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.11 - 2.26i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.23T + 43T^{2} \)
47 \( 1 + (-3.38 + 10.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.572 + 1.76i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.85 - 4.25i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.881 + 0.640i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.618 - 1.90i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-4.76 + 14.6i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.736 + 0.534i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.76 - 8.50i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.47 + 10.6i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (10.1 - 7.38i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (1.35 - 4.16i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.596827121350262888869175586742, −8.782820609228353372222793504557, −7.37435346051783833747106973541, −6.95822489004390642192067228286, −6.38355728643636588292683341072, −5.52552033387196768701627012912, −3.55462545188999796469619676927, −2.76395082837101238261789359421, −2.31972187279437830230097182537, −0.42766649930402991307225993837, 2.75152473530573359711282191401, 3.46062536943698374850762181972, 4.28188379489853860955154850403, 5.26802470600150847206386781418, 5.82028260854336495542140358607, 7.02781405825948032537981665766, 8.345730889213185954379575284255, 9.074954835540881726658845948715, 9.556360714811055261824482521996, 10.22273304220348085012747637131

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