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Properties

Label 2-950-25.16-c1-0-10
Degree $2$
Conductor $950$
Sign $-0.851 - 0.523i$
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 + (0.401 + 1.23i)3-s + (0.309 + 0.951i)4-s + (1.49 + 1.66i)5-s + (−0.401 + 1.23i)6-s − 2.35·7-s + (−0.309 + 0.951i)8-s + (1.06 − 0.772i)9-s + (0.226 + 2.22i)10-s + (−1.33 − 0.966i)11-s + (−1.05 + 0.763i)12-s + (−4.20 + 3.05i)13-s + (−1.90 − 1.38i)14-s + (−1.46 + 2.50i)15-s + (−0.809 + 0.587i)16-s + (−1.59 + 4.91i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.231 + 0.712i)3-s + (0.154 + 0.475i)4-s + (0.666 + 0.745i)5-s + (−0.163 + 0.504i)6-s − 0.890·7-s + (−0.109 + 0.336i)8-s + (0.354 − 0.257i)9-s + (0.0714 + 0.703i)10-s + (−0.401 − 0.291i)11-s + (−0.303 + 0.220i)12-s + (−1.16 + 0.847i)13-s + (−0.509 − 0.370i)14-s + (−0.377 + 0.647i)15-s + (−0.202 + 0.146i)16-s + (−0.387 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.851 - 0.523i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.851 - 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.573672 + 2.02764i\)
\(L(\frac12)\) \(\approx\) \(0.573672 + 2.02764i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.49 - 1.66i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-0.401 - 1.23i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 2.35T + 7T^{2} \)
11 \( 1 + (1.33 + 0.966i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.20 - 3.05i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.59 - 4.91i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (-3.21 - 2.33i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.627 + 1.93i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.781 - 2.40i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.0385 + 0.0279i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.70 + 6.32i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.73T + 43T^{2} \)
47 \( 1 + (2.25 + 6.94i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.75 - 8.47i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.677 - 0.492i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.747 - 0.543i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.91 + 8.98i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.43 + 10.5i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.38 - 1.73i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.60 - 11.0i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.00 + 6.15i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-12.5 - 9.11i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.84 - 8.76i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.36667257354586527405844457452, −9.455050040911704832498044134858, −9.053898350215705813101335116946, −7.58058521835649356305483984590, −6.82507535710331929726917177070, −6.16310038508218879881884325466, −5.15385654699297704738309236893, −4.09262917884539483591762933165, −3.26512586546544989574423183614, −2.24554259694166215792635688324, 0.76590518616052884654094331669, 2.25611115742749664700846539613, 2.89461623960843040778737408658, 4.53645191753598811299992620374, 5.17312513527341525435523303544, 6.19353461189741845682978567108, 7.12271686043010424180805505181, 7.83864830712602632048498362354, 9.095395176776775368500139578451, 9.791554815233619807656271551654

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