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Properties

Label 2-950-19.6-c1-0-2
Degree $2$
Conductor $950$
Sign $-0.894 - 0.447i$
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 + (−3.03 − 1.10i)3-s + (0.173 − 0.984i)4-s + (3.03 − 1.10i)6-s + (−1.36 + 2.35i)7-s + (0.500 + 0.866i)8-s + (5.67 + 4.75i)9-s + (−1.18 − 2.05i)11-s + (−1.61 + 2.79i)12-s + (−0.0945 + 0.0344i)13-s + (−0.472 − 2.68i)14-s + (−0.939 − 0.342i)16-s + (4.17 − 3.50i)17-s − 7.40·18-s + (1.18 − 4.19i)19-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−1.74 − 0.636i)3-s + (0.0868 − 0.492i)4-s + (1.23 − 0.450i)6-s + (−0.514 + 0.891i)7-s + (0.176 + 0.306i)8-s + (1.89 + 1.58i)9-s + (−0.358 − 0.621i)11-s + (−0.465 + 0.806i)12-s + (−0.0262 + 0.00954i)13-s + (−0.126 − 0.716i)14-s + (−0.234 − 0.0855i)16-s + (1.01 − 0.849i)17-s − 1.74·18-s + (0.271 − 0.962i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0277778 + 0.117542i\)
\(L(\frac12)\) \(\approx\) \(0.0277778 + 0.117542i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 + 4.19i)T \)
good3 \( 1 + (3.03 + 1.10i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (1.36 - 2.35i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.18 + 2.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0945 - 0.0344i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-4.17 + 3.50i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.235 + 1.33i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.61 + 1.35i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (5.26 - 9.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.18T + 37T^{2} \)
41 \( 1 + (-3.30 - 1.20i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.22 + 6.96i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-6.13 - 5.14i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.94 - 11.0i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (9.13 - 7.66i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.0303 - 0.172i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (4.17 + 3.50i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.240 - 1.36i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (11.3 + 4.13i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (3.51 + 1.27i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (4.50 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.26 - 2.64i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (2.26 - 1.89i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.61723601824485771086807791577, −9.566881885684059807371056987184, −8.764417975209456845220726762169, −7.49348093875335871159146575052, −7.03882276008923091604798686141, −5.95255025235329505303912224227, −5.62020023606650532377610862875, −4.76578457708997877038667273573, −2.80186518456344842543237928360, −1.17832547423058344300627740520, 0.10213083774749937611294792406, 1.47178249880389184174522292714, 3.57919246306295846253500570596, 4.21743283970460829498927267469, 5.39775977680271833468683005568, 6.12605588129847766277786212578, 7.14439102939447481966179049993, 7.82700089250711738448106348135, 9.374132075628066012960287380968, 10.08630691870059414281840545777

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