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Properties

Label 2-195-65.4-c1-0-5
Degree $2$
Conductor $195$
Sign $-0.283 - 0.959i$
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  + (−0.946 + 1.63i)2-s + (0.866 + 0.5i)3-s + (−0.790 − 1.36i)4-s + (1.73 − 1.40i)5-s + (−1.63 + 0.946i)6-s + (1.37 + 2.38i)7-s − 0.794·8-s + (0.499 + 0.866i)9-s + (0.667 + 4.17i)10-s + (0.884 + 0.510i)11-s − 1.58i·12-s + (3.58 − 0.416i)13-s − 5.20·14-s + (2.20 − 0.352i)15-s + (2.33 − 4.03i)16-s + (−6.16 + 3.55i)17-s + ⋯
L(s)  = 1  + (−0.668 + 1.15i)2-s + (0.499 + 0.288i)3-s + (−0.395 − 0.684i)4-s + (0.776 − 0.630i)5-s + (−0.668 + 0.386i)6-s + (0.520 + 0.900i)7-s − 0.280·8-s + (0.166 + 0.288i)9-s + (0.211 + 1.32i)10-s + (0.266 + 0.154i)11-s − 0.456i·12-s + (0.993 − 0.115i)13-s − 1.39·14-s + (0.570 − 0.0911i)15-s + (0.582 − 1.00i)16-s + (−1.49 + 0.862i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.283 - 0.959i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.283 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.676407 + 0.904883i\)
\(L(\frac12)\) \(\approx\) \(0.676407 + 0.904883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-1.73 + 1.40i)T \)
13 \( 1 + (-3.58 + 0.416i)T \)
good2 \( 1 + (0.946 - 1.63i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-1.37 - 2.38i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.884 - 0.510i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (6.16 - 3.55i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.78 - 1.03i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.94 + 3.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.43 + 7.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + (-1.77 + 3.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.74 - 1.58i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.52 - 2.03i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.29T + 47T^{2} \)
53 \( 1 + 7.33iT - 53T^{2} \)
59 \( 1 + (-2.57 + 1.48i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.14 + 5.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.76 + 6.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.2 - 5.90i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + 4.42T + 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + (-0.576 - 0.332i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.58 + 7.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.96726103505649124924814989612, −11.89877303276449073602015886152, −10.45663311698804907167225583880, −9.357195080815696664907086310163, −8.505398664770473171036853534825, −8.271775846659665257695838214882, −6.47111738937174260579613667230, −5.81930157392996653384063210983, −4.37689892110627974677035059275, −2.16847586670270945209576987901, 1.44811451936271399331158878765, 2.67817326093490871727588091238, 4.06137396884907718347290013059, 6.12520261705251893985500758114, 7.20425020582306504015338553368, 8.603821539300250448908872652128, 9.342620340833593361238595016710, 10.45004843205033057863213201661, 10.98464542309873179472784105199, 11.89386718276011215823547642662

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