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Properties

Label 2-197-197.14-c2-0-19
Degree $2$
Conductor $197$
Sign $0.516 + 0.856i$
Analytic cond. $5.36786$
Root an. cond. $2.31686$
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  + (−1.85 + 1.85i)2-s + (0.722 − 0.722i)3-s − 2.87i·4-s + (−6.14 + 6.14i)5-s + 2.67i·6-s − 9.45i·7-s + (−2.08 − 2.08i)8-s + 7.95i·9-s − 22.7i·10-s + (3.71 − 3.71i)11-s + (−2.07 − 2.07i)12-s + (0.369 − 0.369i)13-s + (17.5 + 17.5i)14-s + 8.87i·15-s + 19.2·16-s + (−15.3 − 15.3i)17-s + ⋯
L(s)  = 1  + (−0.927 + 0.927i)2-s + (0.240 − 0.240i)3-s − 0.718i·4-s + (−1.22 + 1.22i)5-s + 0.446i·6-s − 1.35i·7-s + (−0.260 − 0.260i)8-s + 0.884i·9-s − 2.27i·10-s + (0.337 − 0.337i)11-s + (−0.172 − 0.172i)12-s + (0.0284 − 0.0284i)13-s + (1.25 + 1.25i)14-s + 0.591i·15-s + 1.20·16-s + (−0.905 − 0.905i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(197\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(5.36786\)
Root analytic conductor: \(2.31686\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 197,\ (\ :1),\ 0.516 + 0.856i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.285630 - 0.161297i\)
\(L(\frac12)\) \(\approx\) \(0.285630 - 0.161297i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 + (-40.5 + 192. i)T \)
good2 \( 1 + (1.85 - 1.85i)T - 4iT^{2} \)
3 \( 1 + (-0.722 + 0.722i)T - 9iT^{2} \)
5 \( 1 + (6.14 - 6.14i)T - 25iT^{2} \)
7 \( 1 + 9.45iT - 49T^{2} \)
11 \( 1 + (-3.71 + 3.71i)T - 121iT^{2} \)
13 \( 1 + (-0.369 + 0.369i)T - 169iT^{2} \)
17 \( 1 + (15.3 + 15.3i)T + 289iT^{2} \)
19 \( 1 + 17.7iT - 361T^{2} \)
23 \( 1 + 4.12T + 529T^{2} \)
29 \( 1 - 10.3T + 841T^{2} \)
31 \( 1 + (-34.3 + 34.3i)T - 961iT^{2} \)
37 \( 1 + 53.7T + 1.36e3T^{2} \)
41 \( 1 - 17.7iT - 1.68e3T^{2} \)
43 \( 1 + 15.8iT - 1.84e3T^{2} \)
47 \( 1 + 59.0iT - 2.20e3T^{2} \)
53 \( 1 + 65.5T + 2.80e3T^{2} \)
59 \( 1 + 1.87T + 3.48e3T^{2} \)
61 \( 1 + 67.6T + 3.72e3T^{2} \)
67 \( 1 + (73.0 - 73.0i)T - 4.48e3iT^{2} \)
71 \( 1 + (77.9 + 77.9i)T + 5.04e3iT^{2} \)
73 \( 1 + (37.2 - 37.2i)T - 5.32e3iT^{2} \)
79 \( 1 + (-4.08 - 4.08i)T + 6.24e3iT^{2} \)
83 \( 1 - 53.8iT - 6.88e3T^{2} \)
89 \( 1 + (-30.5 - 30.5i)T + 7.92e3iT^{2} \)
97 \( 1 - 8.46iT - 9.40e3T^{2} \)
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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

−11.77347964627880178866046710315, −10.91894997940126247277011945088, −10.15273518599908420950657806669, −8.709038233024351034502662910885, −7.76087429340537378345163862728, −7.19250898229744500192070462448, −6.59594355002220569552690438059, −4.39045408513033676815479325435, −3.07052435249348439333356355017, −0.25748343455340493497621127713, 1.52185705508447556633305579656, 3.29365206298301854848716064639, 4.59909787161423990173724261785, 6.10772523401405849903843318087, 8.042915757666918741262309139941, 8.826663457179970565891997794897, 9.105452266901446598399253092094, 10.37239624169984081338405931116, 11.66683505820000969862949590038, 12.19274744806313571496429222000

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