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Properties

Label 2-950-95.8-c1-0-23
Degree $2$
Conductor $950$
Sign $0.606 + 0.794i$
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  + (0.965 − 0.258i)2-s + (0.0641 + 0.239i)3-s + (0.866 − 0.499i)4-s + (0.123 + 0.214i)6-s + (2.17 − 2.17i)7-s + (0.707 − 0.707i)8-s + (2.54 − 1.46i)9-s − 0.518·11-s + (0.175 + 0.175i)12-s + (−4.17 − 1.11i)13-s + (1.53 − 2.66i)14-s + (0.500 − 0.866i)16-s + (−0.366 − 1.36i)17-s + (2.07 − 2.07i)18-s + (3.08 + 3.08i)19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.0370 + 0.138i)3-s + (0.433 − 0.249i)4-s + (0.0505 + 0.0875i)6-s + (0.822 − 0.822i)7-s + (0.249 − 0.249i)8-s + (0.848 − 0.489i)9-s − 0.156·11-s + (0.0505 + 0.0505i)12-s + (−1.15 − 0.310i)13-s + (0.411 − 0.712i)14-s + (0.125 − 0.216i)16-s + (−0.0889 − 0.332i)17-s + (0.489 − 0.489i)18-s + (0.707 + 0.706i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.606 + 0.794i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.606 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38735 - 1.18095i\)
\(L(\frac12)\) \(\approx\) \(2.38735 - 1.18095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
19 \( 1 + (-3.08 - 3.08i)T \)
good3 \( 1 + (-0.0641 - 0.239i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-2.17 + 2.17i)T - 7iT^{2} \)
11 \( 1 + 0.518T + 11T^{2} \)
13 \( 1 + (4.17 + 1.11i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.366 + 1.36i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (-0.343 + 1.28i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.52 + 6.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.22iT - 31T^{2} \)
37 \( 1 + (-7.09 - 7.09i)T + 37iT^{2} \)
41 \( 1 + (2.55 + 1.47i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.14 - 2.18i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-10.0 - 2.70i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-9.32 - 2.49i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.414 + 0.718i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.88 - 5.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.922 + 3.44i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (9.71 + 5.60i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.87 + 0.501i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.10 + 5.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.80 - 5.80i)T + 83iT^{2} \)
89 \( 1 + (-2.05 - 3.55i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.97 - 1.86i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.06713203298517088217651889824, −9.375686169269053280676643837930, −7.86663877210826625837690834325, −7.47261392352033809929280055794, −6.47968386788763141103702649532, −5.31091321335891337727090342662, −4.52423156599309716080390902230, −3.79454116151109291141884030765, −2.48297477120975027833417849962, −1.10319402065817391364689625997, 1.75774354386558615672103747929, 2.66033279627158949559164318700, 4.08956061449730867670465483627, 5.05940438402166117086366113686, 5.49288680303090848105131089051, 6.94754934102277428206083496530, 7.39563002877639363553596344236, 8.365959874113441953673643772176, 9.279285289890854170483056950893, 10.24423209208875133628476346934

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