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Properties

Label 2-950-5.4-c3-0-22
Degree $2$
Conductor $950$
Sign $-0.894 - 0.447i$
Analytic cond. $56.0518$
Root an. cond. $7.48677$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 0.227i·3-s − 4·4-s + 0.455·6-s + 8.08i·7-s − 8i·8-s + 26.9·9-s − 12.7·11-s + 0.911i·12-s + 47.0i·13-s − 16.1·14-s + 16·16-s − 31.4i·17-s + 53.8i·18-s − 19·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.0438i·3-s − 0.5·4-s + 0.0310·6-s + 0.436i·7-s − 0.353i·8-s + 0.998·9-s − 0.350·11-s + 0.0219i·12-s + 1.00i·13-s − 0.308·14-s + 0.250·16-s − 0.448i·17-s + 0.705i·18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(56.0518\)
Root analytic conductor: \(7.48677\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.486628999\)
\(L(\frac12)\) \(\approx\) \(1.486628999\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 \)
19 \( 1 + 19T \)
good3 \( 1 + 0.227iT - 27T^{2} \)
7 \( 1 - 8.08iT - 343T^{2} \)
11 \( 1 + 12.7T + 1.33e3T^{2} \)
13 \( 1 - 47.0iT - 2.19e3T^{2} \)
17 \( 1 + 31.4iT - 4.91e3T^{2} \)
23 \( 1 - 19.0iT - 1.21e4T^{2} \)
29 \( 1 + 91.2T + 2.43e4T^{2} \)
31 \( 1 - 293.T + 2.97e4T^{2} \)
37 \( 1 - 215. iT - 5.06e4T^{2} \)
41 \( 1 + 67.7T + 6.89e4T^{2} \)
43 \( 1 + 308. iT - 7.95e4T^{2} \)
47 \( 1 - 108. iT - 1.03e5T^{2} \)
53 \( 1 - 682. iT - 1.48e5T^{2} \)
59 \( 1 - 250.T + 2.05e5T^{2} \)
61 \( 1 + 317.T + 2.26e5T^{2} \)
67 \( 1 - 940. iT - 3.00e5T^{2} \)
71 \( 1 + 395.T + 3.57e5T^{2} \)
73 \( 1 + 975. iT - 3.89e5T^{2} \)
79 \( 1 + 922.T + 4.93e5T^{2} \)
83 \( 1 - 1.16e3iT - 5.71e5T^{2} \)
89 \( 1 + 685.T + 7.04e5T^{2} \)
97 \( 1 - 211. iT - 9.12e5T^{2} \)
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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

−9.834247491382805849231990361517, −9.118803812754572001143266344259, −8.291370791244383718228070725198, −7.34584777628186579489373444316, −6.72468612745487012113265938674, −5.79688087119556351076409732020, −4.76308627937585315397807461318, −4.05188864503611064085437533567, −2.59746863933298686898771608305, −1.25674341808886968829158077563, 0.40796535473073978833393091050, 1.54645259115709705542391750058, 2.78505262431227025441049821529, 3.85677621974777859764350776479, 4.65111883280595017592600939884, 5.68210666326619968666826511345, 6.81245222323688551714160244879, 7.76963253262274756881055656989, 8.456725669561260600204023158721, 9.615224267394108591655368515689

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