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Properties

Label 2-95-19.7-c3-0-5
Degree $2$
Conductor $95$
Sign $-0.952 - 0.304i$
Analytic cond. $5.60518$
Root an. cond. $2.36752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.40 + 4.15i)2-s + (1.64 + 2.84i)3-s + (−7.53 + 13.0i)4-s + (2.5 + 4.33i)5-s + (−7.89 + 13.6i)6-s − 11.3·7-s − 33.9·8-s + (8.10 − 14.0i)9-s + (−12.0 + 20.7i)10-s + 22.0·11-s − 49.5·12-s + (3.78 − 6.55i)13-s + (−27.2 − 47.1i)14-s + (−8.21 + 14.2i)15-s + (−21.2 − 36.8i)16-s + (40.9 + 70.9i)17-s + ⋯
L(s)  = 1  + (0.849 + 1.47i)2-s + (0.316 + 0.547i)3-s + (−0.942 + 1.63i)4-s + (0.223 + 0.387i)5-s + (−0.536 + 0.929i)6-s − 0.611·7-s − 1.50·8-s + (0.300 − 0.519i)9-s + (−0.379 + 0.657i)10-s + 0.604·11-s − 1.19·12-s + (0.0807 − 0.139i)13-s + (−0.519 − 0.899i)14-s + (−0.141 + 0.244i)15-s + (−0.332 − 0.576i)16-s + (0.584 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.952 - 0.304i$
Analytic conductor: \(5.60518\)
Root analytic conductor: \(2.36752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :3/2),\ -0.952 - 0.304i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.382876 + 2.45556i\)
\(L(\frac12)\) \(\approx\) \(0.382876 + 2.45556i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.5 - 4.33i)T \)
19 \( 1 + (82.4 + 7.97i)T \)
good2 \( 1 + (-2.40 - 4.15i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-1.64 - 2.84i)T + (-13.5 + 23.3i)T^{2} \)
7 \( 1 + 11.3T + 343T^{2} \)
11 \( 1 - 22.0T + 1.33e3T^{2} \)
13 \( 1 + (-3.78 + 6.55i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-40.9 - 70.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-76.7 + 132. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-15.8 + 27.4i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 51.8T + 2.97e4T^{2} \)
37 \( 1 - 66.6T + 5.06e4T^{2} \)
41 \( 1 + (-121. - 209. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-143. - 248. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-159. + 275. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-108. + 188. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (275. + 477. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-91.2 + 158. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-417. + 722. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (149. + 258. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (304. + 526. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (124. + 214. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.19e3T + 5.71e5T^{2} \)
89 \( 1 + (-343. + 594. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-396. - 686. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.59888546821704214364302711307, −13.13896952868594464549584231073, −12.46062496422427848222145396986, −10.58867007250602309944441057827, −9.343263574422274996911339137031, −8.218763021480389694373672339918, −6.73251918473809794329623912705, −6.13938273649481981279542674558, −4.48201084120380200608372206134, −3.42057592393688813222131978530, 1.28409319467531454437135292061, 2.69406481991469485120454004281, 4.16028475839509612241927240308, 5.54186853618494731996376588878, 7.23004388471215829265933050177, 9.007171966546100244798121371387, 9.988196039852682221366472159196, 11.12361898364166511336399513238, 12.21644711877008451065474090491, 12.94051427235809595285236487820

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