Properties

Label 2-950-95.27-c1-0-18
Degree $2$
Conductor $950$
Sign $0.566 - 0.823i$
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 + (1.93 − 0.517i)3-s + (−0.866 + 0.499i)4-s + (0.999 + 1.73i)6-s + (1.22 + 1.22i)7-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + 3·11-s + (−1.41 + 1.41i)12-s + (0.517 − 1.93i)13-s + (−0.866 + 1.49i)14-s + (0.500 − 0.866i)16-s + (0.707 + 0.707i)18-s + (2.59 + 3.5i)19-s + (2.99 + 1.73i)21-s + (0.776 + 2.89i)22-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (1.11 − 0.298i)3-s + (−0.433 + 0.249i)4-s + (0.408 + 0.707i)6-s + (0.462 + 0.462i)7-s + (−0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + 0.904·11-s + (−0.408 + 0.408i)12-s + (0.143 − 0.535i)13-s + (−0.231 + 0.400i)14-s + (0.125 − 0.216i)16-s + (0.166 + 0.166i)18-s + (0.596 + 0.802i)19-s + (0.654 + 0.377i)21-s + (0.165 + 0.617i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.566 - 0.823i$
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: \((2,\ 950,\ (\ :1/2),\ 0.566 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31732 + 1.21863i\)
\(L(\frac12)\) \(\approx\) \(2.31732 + 1.21863i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 3.5i)T \)
good3 \( 1 + (-1.93 + 0.517i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.22 - 1.22i)T + 7iT^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-0.517 + 1.93i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-5.01 - 1.34i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (0.707 - 0.707i)T - 37iT^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.79 - 6.69i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.68 - 10.0i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.776 + 2.89i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.19 + 9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.93 - 0.517i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (9 + 5.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.896 - 3.34i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.73 + 3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-2.59 - 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.07 + 7.72i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.737468034929108163279185426656, −9.126655711765166865592920945648, −8.321419347407784485466645924727, −7.82762440685644552946166085514, −6.92220532847247057354473356580, −5.89832635170620984305690852578, −5.00900286895718197541014256292, −3.76708816936639733019005405866, −2.91837082323798011854394443062, −1.54212911028611251017398840878, 1.25341990935569037216164440887, 2.52788373168010515397312501735, 3.50620816272029913740613931142, 4.25053450862243553975853912896, 5.21878091003471359974937932247, 6.62938200974358829435437946637, 7.49387403324065479289053401669, 8.739794639728414282791439702512, 8.929500309133095693877799753593, 9.844826329961150683126540792812

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