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Properties

Label 2-95-19.16-c1-0-3
Degree $2$
Conductor $95$
Sign $0.993 + 0.115i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + (0.528 + 0.443i)2-s + (0.652 − 0.237i)3-s + (−0.264 − 1.50i)4-s + (0.173 − 0.984i)5-s + (0.450 + 0.164i)6-s + (1.16 + 2.02i)7-s + (1.21 − 2.10i)8-s + (−1.92 + 1.61i)9-s + (0.528 − 0.443i)10-s + (−2.28 + 3.96i)11-s + (−0.529 − 0.916i)12-s + (−1.20 − 0.438i)13-s + (−0.279 + 1.58i)14-s + (−0.120 − 0.684i)15-s + (−1.28 + 0.468i)16-s + (−0.501 − 0.420i)17-s + ⋯
L(s)  = 1  + (0.373 + 0.313i)2-s + (0.376 − 0.137i)3-s + (−0.132 − 0.750i)4-s + (0.0776 − 0.440i)5-s + (0.183 + 0.0669i)6-s + (0.441 + 0.764i)7-s + (0.429 − 0.744i)8-s + (−0.642 + 0.539i)9-s + (0.167 − 0.140i)10-s + (−0.690 + 1.19i)11-s + (−0.152 − 0.264i)12-s + (−0.333 − 0.121i)13-s + (−0.0747 + 0.424i)14-s + (−0.0311 − 0.176i)15-s + (−0.321 + 0.117i)16-s + (−0.121 − 0.102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.993 + 0.115i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.993 + 0.115i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25192 - 0.0722573i\)
\(L(\frac12)\) \(\approx\) \(1.25192 - 0.0722573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-3.67 + 2.34i)T \)
good2 \( 1 + (-0.528 - 0.443i)T + (0.347 + 1.96i)T^{2} \)
3 \( 1 + (-0.652 + 0.237i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.16 - 2.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.28 - 3.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.20 + 0.438i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.501 + 0.420i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.966 + 5.48i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (3.62 - 3.04i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.24 - 3.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.79T + 37T^{2} \)
41 \( 1 + (8.17 - 2.97i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.66 + 9.44i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-4.84 + 4.06i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-1.14 - 6.50i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-4.51 - 3.78i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.30 + 7.38i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-10.0 + 8.39i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.651 + 3.69i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (7.48 - 2.72i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (5.92 - 2.15i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-4.91 - 8.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (11.4 + 4.16i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-3.22 - 2.70i)T + (16.8 + 95.5i)T^{2} \)
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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

−14.07678871478744155261076836279, −13.13108354112366640442271820120, −12.07158768313527822998298351376, −10.68285867815505474257413162938, −9.546682720853121976406333954850, −8.446221767059002592537125169903, −7.14706907505468502393786420292, −5.48425107548733331343300915306, −4.80345278759043938481512574443, −2.26689143889831440333173002985, 2.89345669745828864413365376674, 3.96131752053887658939684312140, 5.70375312288071039060043918451, 7.52806396893408812164873273373, 8.299492392156813760997806031197, 9.707926158206654778125534595068, 11.15301680866560580634443944975, 11.70642711052697535938583288809, 13.24663224758600584015422540933, 13.85537494297990048038962065751

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