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Properties

Label 2-950-19.16-c1-0-24
Degree $2$
Conductor $950$
Sign $0.944 - 0.328i$
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  + (0.766 + 0.642i)2-s + (2.55 − 0.931i)3-s + (0.173 + 0.984i)4-s + (2.55 + 0.931i)6-s + (1.33 + 2.31i)7-s + (−0.500 + 0.866i)8-s + (3.37 − 2.83i)9-s + (2.46 − 4.26i)11-s + (1.36 + 2.35i)12-s + (−4.24 − 1.54i)13-s + (−0.463 + 2.62i)14-s + (−0.939 + 0.342i)16-s + (3.76 + 3.15i)17-s + 4.41·18-s + (−0.628 − 4.31i)19-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (1.47 − 0.537i)3-s + (0.0868 + 0.492i)4-s + (1.04 + 0.380i)6-s + (0.504 + 0.873i)7-s + (−0.176 + 0.306i)8-s + (1.12 − 0.945i)9-s + (0.742 − 1.28i)11-s + (0.392 + 0.680i)12-s + (−1.17 − 0.428i)13-s + (−0.123 + 0.702i)14-s + (−0.234 + 0.0855i)16-s + (0.911 + 0.765i)17-s + 1.03·18-s + (−0.144 − 0.989i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.944 - 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.56877 + 0.602228i\)
\(L(\frac12)\) \(\approx\) \(3.56877 + 0.602228i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 \)
19 \( 1 + (0.628 + 4.31i)T \)
good3 \( 1 + (-2.55 + 0.931i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.33 - 2.31i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.46 + 4.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.24 + 1.54i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.76 - 3.15i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.542 - 3.07i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.227 + 0.190i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-3.13 - 5.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.79T + 37T^{2} \)
41 \( 1 + (7.15 - 2.60i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.528 - 2.99i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.73 + 1.45i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-2.29 - 13.0i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (9.63 + 8.08i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.806 + 4.57i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.47 + 6.27i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.702 + 3.98i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (4.45 - 1.62i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (0.708 - 0.257i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (5.52 + 9.57i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.44 + 2.71i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (5.45 + 4.57i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.761658041011785329374468677311, −8.796209575304858691811925574811, −8.460671315843248316987005063669, −7.66144860142217391241297517472, −6.82548706456779862559178134655, −5.77788274891618669008492615819, −4.84917469823153785438161398848, −3.42746557078379538151635276384, −2.90662152141922221972336223287, −1.66533854261288871104976390139, 1.65462195134247451348145213499, 2.59491914952771027974907829570, 3.78635304530829748598488659348, 4.33003677045207297633467936373, 5.16910883432000433425765447036, 6.89340320958854158068163031685, 7.46842918984798583172175773165, 8.392325837691130224637179323204, 9.449450025496232033130504961324, 9.942585910110435295368538832912

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