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Properties

Label 2-950-95.27-c1-0-20
Degree $2$
Conductor $950$
Sign $0.707 + 0.706i$
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  + (−0.258 − 0.965i)2-s + (2.53 − 0.680i)3-s + (−0.866 + 0.499i)4-s + (−1.31 − 2.27i)6-s + (2.47 + 2.47i)7-s + (0.707 + 0.707i)8-s + (3.38 − 1.95i)9-s + 0.295·11-s + (−1.85 + 1.85i)12-s + (−0.347 + 1.29i)13-s + (1.75 − 3.03i)14-s + (0.500 − 0.866i)16-s + (0.682 − 0.182i)17-s + (−2.76 − 2.76i)18-s + (−1.91 − 3.91i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (1.46 − 0.392i)3-s + (−0.433 + 0.249i)4-s + (−0.536 − 0.929i)6-s + (0.936 + 0.936i)7-s + (0.249 + 0.249i)8-s + (1.12 − 0.651i)9-s + 0.0891·11-s + (−0.536 + 0.536i)12-s + (−0.0964 + 0.360i)13-s + (0.468 − 0.810i)14-s + (0.125 − 0.216i)16-s + (0.165 − 0.0443i)17-s + (−0.651 − 0.651i)18-s + (−0.440 − 0.897i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.707 + 0.706i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.707 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33842 - 0.968042i\)
\(L(\frac12)\) \(\approx\) \(2.33842 - 0.968042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
19 \( 1 + (1.91 + 3.91i)T \)
good3 \( 1 + (-2.53 + 0.680i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.47 - 2.47i)T + 7iT^{2} \)
11 \( 1 - 0.295T + 11T^{2} \)
13 \( 1 + (0.347 - 1.29i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.682 + 0.182i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-7.49 - 2.00i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.37 + 4.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.65iT - 31T^{2} \)
37 \( 1 + (-2.70 + 2.70i)T - 37iT^{2} \)
41 \( 1 + (-7.26 - 4.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.60 + 9.73i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.07 - 7.72i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.590 - 2.20i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-6.62 + 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.50 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.20 + 2.19i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.84 + 2.21i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.82 + 6.80i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.58 - 14.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.4 - 12.4i)T - 83iT^{2} \)
89 \( 1 + (0.137 + 0.237i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.51 - 5.63i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.533286088006723626024440353629, −9.091516355900353631288655648820, −8.457665552270630380555548549015, −7.75555344488951872557098817510, −6.84960430543352166584731288370, −5.35270901621985527576854999254, −4.38786906871021552318950770674, −3.14873963565084342690156338026, −2.40491944444709668734620499119, −1.49589817401266155720305937416, 1.37776656417950376712145256807, 2.84988014607017737000921092932, 3.99871714435283713640123303182, 4.62803969463901070707943118739, 5.84941916595592377701591927617, 7.23514496778671986913633758780, 7.65890327373768999515637866098, 8.478311187030411753964275065593, 9.020589394516906431190951700378, 10.01562713790858910445068025647

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