Properties

Label 2-950-475.117-c1-0-11
Degree $2$
Conductor $950$
Sign $-0.786 + 0.618i$
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  + (0.390 + 0.920i)2-s + (−1.09 + 0.213i)3-s + (−0.694 + 0.719i)4-s + (1.58 + 1.58i)5-s + (−0.626 − 0.928i)6-s + (−1.16 + 4.34i)7-s + (−0.933 − 0.358i)8-s + (−1.61 + 0.654i)9-s + (−0.837 + 2.07i)10-s + (−0.164 − 0.0349i)11-s + (0.609 − 0.938i)12-s + (−3.94 + 0.483i)13-s + (−4.45 + 0.626i)14-s + (−2.07 − 1.39i)15-s + (−0.0348 − 0.999i)16-s + (3.61 − 6.01i)17-s + ⋯
L(s)  = 1  + (0.276 + 0.650i)2-s + (−0.634 + 0.123i)3-s + (−0.347 + 0.359i)4-s + (0.707 + 0.707i)5-s + (−0.255 − 0.378i)6-s + (−0.440 + 1.64i)7-s + (−0.330 − 0.126i)8-s + (−0.539 + 0.218i)9-s + (−0.264 + 0.655i)10-s + (−0.0495 − 0.0105i)11-s + (0.176 − 0.271i)12-s + (−1.09 + 0.134i)13-s + (−1.19 + 0.167i)14-s + (−0.535 − 0.361i)15-s + (−0.00872 − 0.249i)16-s + (0.876 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.786 + 0.618i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.786 + 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270721 - 0.782256i\)
\(L(\frac12)\) \(\approx\) \(0.270721 - 0.782256i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.390 - 0.920i)T \)
5 \( 1 + (-1.58 - 1.58i)T \)
19 \( 1 + (-0.268 - 4.35i)T \)
good3 \( 1 + (1.09 - 0.213i)T + (2.78 - 1.12i)T^{2} \)
7 \( 1 + (1.16 - 4.34i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.164 + 0.0349i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (3.94 - 0.483i)T + (12.6 - 3.14i)T^{2} \)
17 \( 1 + (-3.61 + 6.01i)T + (-7.98 - 15.0i)T^{2} \)
23 \( 1 + (-1.26 + 4.14i)T + (-19.0 - 12.8i)T^{2} \)
29 \( 1 + (-0.135 + 0.541i)T + (-25.6 - 13.6i)T^{2} \)
31 \( 1 + (5.53 + 0.581i)T + (30.3 + 6.44i)T^{2} \)
37 \( 1 + (0.541 + 1.06i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (3.66 - 0.127i)T + (40.9 - 2.86i)T^{2} \)
43 \( 1 + (-6.71 - 9.59i)T + (-14.7 + 40.4i)T^{2} \)
47 \( 1 + (-5.05 + 3.03i)T + (22.0 - 41.4i)T^{2} \)
53 \( 1 + (0.0940 + 5.39i)T + (-52.9 + 1.84i)T^{2} \)
59 \( 1 + (10.1 - 6.37i)T + (25.8 - 53.0i)T^{2} \)
61 \( 1 + (7.83 - 4.16i)T + (34.1 - 50.5i)T^{2} \)
67 \( 1 + (2.40 - 2.77i)T + (-9.32 - 66.3i)T^{2} \)
71 \( 1 + (-3.02 + 1.47i)T + (43.7 - 55.9i)T^{2} \)
73 \( 1 + (-6.12 - 0.751i)T + (70.8 + 17.6i)T^{2} \)
79 \( 1 + (5.38 - 7.98i)T + (-29.5 - 73.2i)T^{2} \)
83 \( 1 + (6.20 + 7.66i)T + (-17.2 + 81.1i)T^{2} \)
89 \( 1 + (0.0191 - 0.549i)T + (-88.7 - 6.20i)T^{2} \)
97 \( 1 + (2.42 - 2.10i)T + (13.4 - 96.0i)T^{2} \)
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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

−10.43920083614431403774987624012, −9.569898357896295876628805705551, −9.039048937132393095480203377593, −7.87902095475151367091664032533, −6.95422047057239348029677367224, −5.98927143858892484999624762720, −5.59037013608766401314131430211, −4.87227842006768809444097825837, −3.06234602012349533891467281282, −2.42504379083163272022570773216, 0.37593755240548243483932803608, 1.52727309413085629145905575096, 3.11118213227703134150726869034, 4.15699517912432248638107207873, 5.13681976764207129126323533089, 5.85292306621939998663863269974, 6.85703840019091482021566754452, 7.77498249675689706396527830236, 9.036189549538582214235311949109, 9.705041122840471052371038980105

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