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Properties

Label 2-950-95.9-c1-0-25
Degree $2$
Conductor $950$
Sign $0.210 + 0.977i$
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  + (0.984 − 0.173i)2-s + (0.223 + 0.266i)3-s + (0.939 − 0.342i)4-s + (0.266 + 0.223i)6-s + (−1.52 − 0.879i)7-s + (0.866 − 0.5i)8-s + (0.5 − 2.83i)9-s + (−2.11 − 3.66i)11-s + (0.300 + 0.173i)12-s + (−0.684 + 0.815i)13-s + (−1.65 − 0.601i)14-s + (0.766 − 0.642i)16-s + (7.02 − 1.23i)17-s − 2.87i·18-s + (−3.93 − 1.86i)19-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (0.128 + 0.153i)3-s + (0.469 − 0.171i)4-s + (0.108 + 0.0911i)6-s + (−0.575 − 0.332i)7-s + (0.306 − 0.176i)8-s + (0.166 − 0.945i)9-s + (−0.637 − 1.10i)11-s + (0.0868 + 0.0501i)12-s + (−0.189 + 0.226i)13-s + (−0.441 − 0.160i)14-s + (0.191 − 0.160i)16-s + (1.70 − 0.300i)17-s − 0.678i·18-s + (−0.903 − 0.427i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.210 + 0.977i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.210 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69382 - 1.36829i\)
\(L(\frac12)\) \(\approx\) \(1.69382 - 1.36829i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
5 \( 1 \)
19 \( 1 + (3.93 + 1.86i)T \)
good3 \( 1 + (-0.223 - 0.266i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (1.52 + 0.879i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.11 + 3.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.684 - 0.815i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-7.02 + 1.23i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (1.28 + 3.53i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.10 - 6.27i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.41 + 7.64i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.45iT - 37T^{2} \)
41 \( 1 + (-1.43 + 1.20i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.26 + 3.47i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-3.61 - 0.638i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-3.38 - 9.29i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.26 - 7.18i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.98 - 1.81i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-11.4 - 2.02i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.65 + 0.965i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-0.509 - 0.607i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-5.12 + 4.30i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.30 + 0.754i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.12 + 7.65i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.85 + 0.326i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.05665420079570177654441577171, −9.109820417144876584691114066690, −8.164254819695776617193569832570, −7.14558056763069048802324685450, −6.27703995827302933812311486625, −5.57519412430735734252427698880, −4.37591163560444844793102002929, −3.46541127380103588657368424504, −2.71460686469808871254820351602, −0.799693158013389479489915907756, 1.86851081302233195967276320604, 2.84197394023099063371516843951, 3.99691555356826047371386404903, 5.07066461234625838619673557255, 5.73760202727890273440093211720, 6.79693239116784711572880699099, 7.74240994263537824898772002065, 8.156788326267050508288624131527, 9.665315051244995707246565924081, 10.16574315200927941669967256148

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