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Properties

Label 2-950-95.74-c1-0-12
Degree $2$
Conductor $950$
Sign $-0.0218 - 0.999i$
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  + (0.984 + 0.173i)2-s + (−1.07 + 1.28i)3-s + (0.939 + 0.342i)4-s + (−1.28 + 1.07i)6-s + (2.14 − 1.23i)7-s + (0.866 + 0.5i)8-s + (0.0320 + 0.181i)9-s + (−1.45 + 2.51i)11-s + (−1.45 + 0.838i)12-s + (0.833 + 0.993i)13-s + (2.32 − 0.846i)14-s + (0.766 + 0.642i)16-s + (−1.22 − 0.215i)17-s + 0.184i·18-s + (4.22 + 1.08i)19-s + ⋯
L(s)  = 1  + (0.696 + 0.122i)2-s + (−0.622 + 0.742i)3-s + (0.469 + 0.171i)4-s + (−0.524 + 0.440i)6-s + (0.810 − 0.467i)7-s + (0.306 + 0.176i)8-s + (0.0106 + 0.0606i)9-s + (−0.437 + 0.757i)11-s + (−0.419 + 0.242i)12-s + (0.231 + 0.275i)13-s + (0.621 − 0.226i)14-s + (0.191 + 0.160i)16-s + (−0.296 − 0.0523i)17-s + 0.0435i·18-s + (0.968 + 0.248i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0218 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0218 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0218 - 0.999i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0218 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44526 + 1.47725i\)
\(L(\frac12)\) \(\approx\) \(1.44526 + 1.47725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 - 0.173i)T \)
5 \( 1 \)
19 \( 1 + (-4.22 - 1.08i)T \)
good3 \( 1 + (1.07 - 1.28i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-2.14 + 1.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.45 - 2.51i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.833 - 0.993i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.22 + 0.215i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.750 + 2.06i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.12 - 6.38i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.04 - 5.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.44iT - 37T^{2} \)
41 \( 1 + (-1.82 - 1.53i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-3.14 - 8.64i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (10.9 - 1.93i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-4.11 + 11.3i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.351 + 1.99i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-1.25 - 0.458i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-3.31 + 0.584i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (14.2 - 5.19i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (0.360 - 0.429i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (4.96 + 4.16i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (7.32 - 4.22i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9.11 + 7.64i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-2.66 - 0.469i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.39021976081801025933447291281, −9.735336940194046653553524714909, −8.455505679032052907718698159194, −7.58048885464263796618337429400, −6.78297195606287269153326016926, −5.63260140948539393219888465198, −4.78376739177474414661119219773, −4.46596599928891281988283075575, −3.13766166475346821258418433615, −1.64382562649512006377325109061, 0.889916760605600785267192287177, 2.21934535673057200582672644295, 3.42681279073923425469626042834, 4.68365241756828305309245751512, 5.66173158905128189598000888774, 6.07497451303881610998482090390, 7.21028383132173524442385487501, 7.928147598679843872467874868186, 8.896832000497666595884939229898, 10.02256917499942176769468924785

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