Properties

Label 2-950-95.12-c1-0-0
Degree $2$
Conductor $950$
Sign $-0.712 - 0.702i$
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  + (−0.965 − 0.258i)2-s + (0.206 − 0.770i)3-s + (0.866 + 0.499i)4-s + (−0.398 + 0.690i)6-s + (−0.349 − 0.349i)7-s + (−0.707 − 0.707i)8-s + (2.04 + 1.18i)9-s − 1.21·11-s + (0.563 − 0.563i)12-s + (−6.55 + 1.75i)13-s + (0.246 + 0.427i)14-s + (0.500 + 0.866i)16-s + (−1.57 + 5.86i)17-s + (−1.67 − 1.67i)18-s + (−4.09 − 1.48i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.119 − 0.444i)3-s + (0.433 + 0.249i)4-s + (−0.162 + 0.281i)6-s + (−0.131 − 0.131i)7-s + (−0.249 − 0.249i)8-s + (0.682 + 0.394i)9-s − 0.367·11-s + (0.162 − 0.162i)12-s + (−1.81 + 0.487i)13-s + (0.0659 + 0.114i)14-s + (0.125 + 0.216i)16-s + (−0.381 + 1.42i)17-s + (−0.394 − 0.394i)18-s + (−0.940 − 0.340i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.712 - 0.702i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.712 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0703996 + 0.171677i\)
\(L(\frac12)\) \(\approx\) \(0.0703996 + 0.171677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
19 \( 1 + (4.09 + 1.48i)T \)
good3 \( 1 + (-0.206 + 0.770i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.349 + 0.349i)T + 7iT^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 + (6.55 - 1.75i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.57 - 5.86i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (2.37 + 8.86i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.16 - 5.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.84iT - 31T^{2} \)
37 \( 1 + (-2.41 + 2.41i)T - 37iT^{2} \)
41 \( 1 + (3.55 - 2.05i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.62 + 1.50i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.201 + 0.0539i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (13.1 - 3.51i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.0144 + 0.0250i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.90 + 6.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.56 - 13.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.99 - 3.46i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.53 - 2.01i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.17 - 3.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.36 - 2.36i)T - 83iT^{2} \)
89 \( 1 + (2.10 - 3.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.06 - 0.821i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.37537423249504778345617941520, −9.617006296518893100051736027158, −8.544565811648030099754351160350, −7.987164432180933840988323022987, −6.93381824499640883028354130159, −6.58070126284116033714234511041, −5.01383547710423791926652412393, −4.12168484537425590881454179971, −2.54326606700662942080140632536, −1.80071275713478258710639945602, 0.097513335945661510763461604924, 2.02192710374941392369572691328, 3.15240884549610987075397456520, 4.49048349754755350395006999483, 5.32944923377498385934449005140, 6.46159392051774380715814284192, 7.45766443020189630001509333029, 7.86812786860393738651869626322, 9.233142636733324798840660161474, 9.688722851492027884938224202875

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