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Properties

Label 2-95-5.4-c1-0-5
Degree $2$
Conductor $95$
Sign $0.447 + 0.894i$
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  − i·2-s + 4-s + (−1 − 2i)5-s + 2i·7-s − 3i·8-s + 3·9-s + (−2 + i)10-s − 4·11-s + 2i·13-s + 2·14-s − 16-s + 4i·17-s − 3i·18-s − 19-s + (−1 − 2i)20-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.5·4-s + (−0.447 − 0.894i)5-s + 0.755i·7-s − 1.06i·8-s + 9-s + (−0.632 + 0.316i)10-s − 1.20·11-s + 0.554i·13-s + 0.534·14-s − 0.250·16-s + 0.970i·17-s − 0.707i·18-s − 0.229·19-s + (−0.223 − 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.912474 - 0.563940i\)
\(L(\frac12)\) \(\approx\) \(0.912474 - 0.563940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1 + 2i)T \)
19 \( 1 + T \)
good2 \( 1 + iT - 2T^{2} \)
3 \( 1 - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 18iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−13.28263545317051257194728846540, −12.59797454518394382825034810842, −11.84461485558858694293352140635, −10.68427162877309853596214878850, −9.644089965471797413239084841091, −8.336229919754813584904025088006, −7.08905699037559252250676843716, −5.41075217509758322553840415941, −3.84233590946015501174280589102, −1.91034861424110577446402056257, 2.85187722726909546411289531010, 4.78034076207371224866306495445, 6.50918098564082978345606367110, 7.32669114933864019521989753244, 8.105993347947847675102235226930, 10.25869934003058642049905339703, 10.70752799236204843250226126333, 12.02418282732934495801628845568, 13.33833719146884021116825649847, 14.39378607396707998210468115281

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