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Properties

Label 2-950-25.11-c1-0-7
Degree $2$
Conductor $950$
Sign $-0.973 - 0.229i$
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 + (−0.410 + 1.26i)3-s + (0.309 − 0.951i)4-s + (0.971 + 2.01i)5-s + (−0.410 − 1.26i)6-s − 2.95·7-s + (0.309 + 0.951i)8-s + (0.997 + 0.725i)9-s + (−1.97 − 1.05i)10-s + (2.17 − 1.57i)11-s + (1.07 + 0.781i)12-s + (2.42 + 1.76i)13-s + (2.38 − 1.73i)14-s + (−2.94 + 0.401i)15-s + (−0.809 − 0.587i)16-s + (1.79 + 5.51i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.237 + 0.729i)3-s + (0.154 − 0.475i)4-s + (0.434 + 0.900i)5-s + (−0.167 − 0.516i)6-s − 1.11·7-s + (0.109 + 0.336i)8-s + (0.332 + 0.241i)9-s + (−0.622 − 0.334i)10-s + (0.655 − 0.476i)11-s + (0.310 + 0.225i)12-s + (0.672 + 0.488i)13-s + (0.638 − 0.463i)14-s + (−0.760 + 0.103i)15-s + (−0.202 − 0.146i)16-s + (0.434 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111604 + 0.959497i\)
\(L(\frac12)\) \(\approx\) \(0.111604 + 0.959497i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.971 - 2.01i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.410 - 1.26i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 2.95T + 7T^{2} \)
11 \( 1 + (-2.17 + 1.57i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.42 - 1.76i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.79 - 5.51i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (-0.486 + 0.353i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.389 - 1.19i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.662 + 2.03i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.00 + 1.45i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.91 - 1.38i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.37T + 43T^{2} \)
47 \( 1 + (1.13 - 3.48i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.00 + 9.24i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.91 + 2.11i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (5.04 - 3.66i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.60 - 8.02i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.43 + 4.40i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.61 - 2.62i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.33 + 13.3i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.924 + 2.84i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.242 - 0.176i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (1.78 - 5.48i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.33612872274493296331881695996, −9.646455636295891397230573114116, −9.017717060504520102872177737140, −7.922391402109425889288123956692, −6.82686695889623795432031236297, −6.27570830971410136434613078123, −5.56295008845488353654573291415, −4.04641190033680491498901428632, −3.29813785625323786572493183198, −1.68892381050336727296696349103, 0.59396242149060297403654169504, 1.57690106847782156494834603284, 2.98625382770158973318946004140, 4.13550574400915510852287939959, 5.43494588096893929240911447517, 6.44271794495383099082349145083, 7.06391801621567910600961976952, 8.029434054257303541626440569646, 9.129525100575859845905883824902, 9.519072725764603909576457745696

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