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Properties

Label 2-950-95.64-c1-0-15
Degree $2$
Conductor $950$
Sign $0.999 + 0.0379i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + i·7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + 5·11-s + (1.73 + i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 3i·18-s + (0.5 + 4.33i)19-s + (4.33 − 2.5i)22-s + (0.866 + 0.5i)23-s + 1.99·26-s + (0.866 + 0.499i)28-s + (3 − 5.19i)29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.377i·7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + 1.50·11-s + (0.480 + 0.277i)13-s + (0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.707i·18-s + (0.114 + 0.993i)19-s + (0.923 − 0.533i)22-s + (0.180 + 0.104i)23-s + 0.392·26-s + (0.163 + 0.0944i)28-s + (0.557 − 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.999 + 0.0379i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.999 + 0.0379i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39643 - 0.0454671i\)
\(L(\frac12)\) \(\approx\) \(2.39643 - 0.0454671i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
19 \( 1 + (-0.5 - 4.33i)T \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 11iT - 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.19 - 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.33 - 2.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 + 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (12.1 - 7i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.13895998404196928474934498575, −9.245661705474416673785142636780, −8.461627883055480396664624415690, −7.46663885377775094129555481544, −6.28336990482225290358656533672, −5.79376537031522619107026680369, −4.59143986355492498312811661653, −3.79319029918689674980888800524, −2.59929520428756840606197565197, −1.43388620901607753058240750596, 1.11019688924150693909677833388, 2.94110387727781576997642837003, 3.78491720508854624424084559504, 4.68429170050238004615634909356, 5.84621031809005544773727977612, 6.62924758859714427625287195375, 7.15726394075252059024665525902, 8.561148318474120406395712897310, 8.941044483452758945288194564106, 10.03817646324843222578604389373

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