Properties

Label 2-195-3.2-c2-0-17
Degree $2$
Conductor $195$
Sign $0.581 - 0.813i$
Analytic cond. $5.31336$
Root an. cond. $2.30507$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.858i·2-s + (2.44 + 1.74i)3-s + 3.26·4-s − 2.23i·5-s + (−1.49 + 2.09i)6-s + 2.16·7-s + 6.23i·8-s + (2.92 + 8.51i)9-s + 1.92·10-s − 7.88i·11-s + (7.96 + 5.68i)12-s + 3.60·13-s + 1.86i·14-s + (3.89 − 5.45i)15-s + 7.69·16-s − 3.67i·17-s + ⋯
L(s)  = 1  + 0.429i·2-s + (0.813 + 0.581i)3-s + 0.815·4-s − 0.447i·5-s + (−0.249 + 0.349i)6-s + 0.309·7-s + 0.779i·8-s + (0.324 + 0.945i)9-s + 0.192·10-s − 0.716i·11-s + (0.663 + 0.473i)12-s + 0.277·13-s + 0.133i·14-s + (0.259 − 0.363i)15-s + 0.480·16-s − 0.216i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(5.31336\)
Root analytic conductor: \(2.30507\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1),\ 0.581 - 0.813i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.11968 + 1.09115i\)
\(L(\frac12)\) \(\approx\) \(2.11968 + 1.09115i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.44 - 1.74i)T \)
5 \( 1 + 2.23iT \)
13 \( 1 - 3.60T \)
good2 \( 1 - 0.858iT - 4T^{2} \)
7 \( 1 - 2.16T + 49T^{2} \)
11 \( 1 + 7.88iT - 121T^{2} \)
17 \( 1 + 3.67iT - 289T^{2} \)
19 \( 1 + 20.8T + 361T^{2} \)
23 \( 1 - 6.25iT - 529T^{2} \)
29 \( 1 + 2.29iT - 841T^{2} \)
31 \( 1 + 24.7T + 961T^{2} \)
37 \( 1 + 16.3T + 1.36e3T^{2} \)
41 \( 1 + 18.7iT - 1.68e3T^{2} \)
43 \( 1 - 58.4T + 1.84e3T^{2} \)
47 \( 1 + 6.89iT - 2.20e3T^{2} \)
53 \( 1 + 34.1iT - 2.80e3T^{2} \)
59 \( 1 + 78.6iT - 3.48e3T^{2} \)
61 \( 1 + 98.8T + 3.72e3T^{2} \)
67 \( 1 + 55.1T + 4.48e3T^{2} \)
71 \( 1 + 67.9iT - 5.04e3T^{2} \)
73 \( 1 - 99.2T + 5.32e3T^{2} \)
79 \( 1 - 66.9T + 6.24e3T^{2} \)
83 \( 1 + 138. iT - 6.88e3T^{2} \)
89 \( 1 - 34.4iT - 7.92e3T^{2} \)
97 \( 1 + 3.08T + 9.40e3T^{2} \)
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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

−12.48289989131823388558285564366, −11.20938673741593463566566747784, −10.57981953621688444148075218997, −9.204553992498658522443804978877, −8.357450476622968810489584249885, −7.53480839986193582714471873677, −6.15480175865972170497142259951, −4.94371735608356951347768140462, −3.48937432436440286417444228885, −2.01267010179038103839154068940, 1.66689319852118363570622666306, 2.73235763896436529434170761237, 4.06113847829528517377871042866, 6.16712002983077866636496836437, 7.07162118088066251238287711197, 7.928033477598226384157619620723, 9.157851283217594975143638168509, 10.31453509746488912829147133368, 11.14171605689644072061915999982, 12.28324761436236012601683046677

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