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Properties

Label 2-197-197.14-c2-0-10
Degree $2$
Conductor $197$
Sign $-0.658 - 0.752i$
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  + (−1.11 + 1.11i)2-s + (2.19 − 2.19i)3-s + 1.52i·4-s + (−1.87 + 1.87i)5-s + 4.88i·6-s + 5.07i·7-s + (−6.14 − 6.14i)8-s − 0.630i·9-s − 4.17i·10-s + (−9.24 + 9.24i)11-s + (3.33 + 3.33i)12-s + (2.79 − 2.79i)13-s + (−5.64 − 5.64i)14-s + 8.22i·15-s + 7.60·16-s + (−12.5 − 12.5i)17-s + ⋯
L(s)  = 1  + (−0.556 + 0.556i)2-s + (0.731 − 0.731i)3-s + 0.380i·4-s + (−0.374 + 0.374i)5-s + 0.814i·6-s + 0.724i·7-s + (−0.768 − 0.768i)8-s − 0.0701i·9-s − 0.417i·10-s + (−0.840 + 0.840i)11-s + (0.277 + 0.277i)12-s + (0.214 − 0.214i)13-s + (−0.403 − 0.403i)14-s + 0.548i·15-s + 0.475·16-s + (−0.736 − 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(197\)
Sign: $-0.658 - 0.752i$
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: \((2,\ 197,\ (\ :1),\ -0.658 - 0.752i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.400594 + 0.882946i\)
\(L(\frac12)\) \(\approx\) \(0.400594 + 0.882946i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 + (6.06 - 196. i)T \)
good2 \( 1 + (1.11 - 1.11i)T - 4iT^{2} \)
3 \( 1 + (-2.19 + 2.19i)T - 9iT^{2} \)
5 \( 1 + (1.87 - 1.87i)T - 25iT^{2} \)
7 \( 1 - 5.07iT - 49T^{2} \)
11 \( 1 + (9.24 - 9.24i)T - 121iT^{2} \)
13 \( 1 + (-2.79 + 2.79i)T - 169iT^{2} \)
17 \( 1 + (12.5 + 12.5i)T + 289iT^{2} \)
19 \( 1 - 33.2iT - 361T^{2} \)
23 \( 1 - 30.3T + 529T^{2} \)
29 \( 1 + 36.4T + 841T^{2} \)
31 \( 1 + (5.16 - 5.16i)T - 961iT^{2} \)
37 \( 1 + 42.9T + 1.36e3T^{2} \)
41 \( 1 - 0.230iT - 1.68e3T^{2} \)
43 \( 1 + 70.5iT - 1.84e3T^{2} \)
47 \( 1 - 55.0iT - 2.20e3T^{2} \)
53 \( 1 - 74.5T + 2.80e3T^{2} \)
59 \( 1 - 75.7T + 3.48e3T^{2} \)
61 \( 1 - 83.6T + 3.72e3T^{2} \)
67 \( 1 + (16.2 - 16.2i)T - 4.48e3iT^{2} \)
71 \( 1 + (65.9 + 65.9i)T + 5.04e3iT^{2} \)
73 \( 1 + (-53.0 + 53.0i)T - 5.32e3iT^{2} \)
79 \( 1 + (-39.8 - 39.8i)T + 6.24e3iT^{2} \)
83 \( 1 + 121. iT - 6.88e3T^{2} \)
89 \( 1 + (81.1 + 81.1i)T + 7.92e3iT^{2} \)
97 \( 1 - 168. iT - 9.40e3T^{2} \)
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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

−12.73226401051962115661531045156, −11.86585402531543586525079782322, −10.52817686789098658184311763606, −9.187533702111006800091228675610, −8.449930388144303976735236468298, −7.46855231660282221457369602358, −7.07376441517262045430074110141, −5.42952585662251116690812500243, −3.47494695542059710337963504691, −2.24272951333048672610759400019, 0.59817383931822810092612788912, 2.66169053047255502330082588743, 3.99206395103038069077303644419, 5.25196946762335029852093061605, 6.85496984061652156723382527871, 8.489156494175677675596142870333, 8.869271111878537572164542681326, 9.939666037255036639538501247063, 10.84403394388024860374109907101, 11.39377607880859985183057283987

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