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Properties

Label 2-195-65.47-c1-0-2
Degree $2$
Conductor $195$
Sign $-0.996 - 0.0788i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
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  + 1.97i·2-s + (−0.707 + 0.707i)3-s − 1.88·4-s + (0.644 + 2.14i)5-s + (−1.39 − 1.39i)6-s − 0.616·7-s + 0.222i·8-s − 1.00i·9-s + (−4.22 + 1.27i)10-s + (1.14 − 1.14i)11-s + (1.33 − 1.33i)12-s + (−3.56 + 0.554i)13-s − 1.21i·14-s + (−1.96 − 1.05i)15-s − 4.21·16-s + (0.816 − 0.816i)17-s + ⋯
L(s)  = 1  + 1.39i·2-s + (−0.408 + 0.408i)3-s − 0.943·4-s + (0.288 + 0.957i)5-s + (−0.569 − 0.569i)6-s − 0.233·7-s + 0.0786i·8-s − 0.333i·9-s + (−1.33 + 0.401i)10-s + (0.344 − 0.344i)11-s + (0.385 − 0.385i)12-s + (−0.988 + 0.153i)13-s − 0.324i·14-s + (−0.508 − 0.273i)15-s − 1.05·16-s + (0.197 − 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.996 - 0.0788i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.996 - 0.0788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0401761 + 1.01751i\)
\(L(\frac12)\) \(\approx\) \(0.0401761 + 1.01751i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.644 - 2.14i)T \)
13 \( 1 + (3.56 - 0.554i)T \)
good2 \( 1 - 1.97iT - 2T^{2} \)
7 \( 1 + 0.616T + 7T^{2} \)
11 \( 1 + (-1.14 + 1.14i)T - 11iT^{2} \)
17 \( 1 + (-0.816 + 0.816i)T - 17iT^{2} \)
19 \( 1 + (-4.26 + 4.26i)T - 19iT^{2} \)
23 \( 1 + (-5.27 - 5.27i)T + 23iT^{2} \)
29 \( 1 - 2.25iT - 29T^{2} \)
31 \( 1 + (-4.04 - 4.04i)T + 31iT^{2} \)
37 \( 1 - 3.21T + 37T^{2} \)
41 \( 1 + (1.89 + 1.89i)T + 41iT^{2} \)
43 \( 1 + (-0.687 - 0.687i)T + 43iT^{2} \)
47 \( 1 - 9.07T + 47T^{2} \)
53 \( 1 + (2.76 - 2.76i)T - 53iT^{2} \)
59 \( 1 + (-4.05 - 4.05i)T + 59iT^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 - 1.67iT - 67T^{2} \)
71 \( 1 + (2.76 + 2.76i)T + 71iT^{2} \)
73 \( 1 + 15.5iT - 73T^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + 4.88T + 83T^{2} \)
89 \( 1 + (7.34 + 7.34i)T + 89iT^{2} \)
97 \( 1 + 13.3iT - 97T^{2} \)
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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

−13.41569364893745788403602741367, −11.82396409217152499703964364053, −11.04432735286369182731813587992, −9.813067601125124296534921159384, −8.989584706694201457562499268518, −7.41290080645209707574843527222, −6.89904804131304136987786564281, −5.80404538860245325626130573291, −4.87444191074330832373515751423, −3.03439988815320719715564329329, 1.01918452892204078667525538076, 2.48761622512002871070396476513, 4.18584179744792287658291738467, 5.36098767823339269254307741776, 6.80349180814337634996564495347, 8.196872030337067044294917975403, 9.563767811801425759710218482602, 10.01092661917265398431587089639, 11.29176979525149829767629035729, 12.17003050566015239897186745895

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