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Properties

Label 2-195-39.8-c1-0-8
Degree $2$
Conductor $195$
Sign $-0.227 - 0.973i$
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  + (−1.57 + 1.57i)2-s + (1.73 + 0.0822i)3-s − 2.98i·4-s + (0.707 − 0.707i)5-s + (−2.86 + 2.60i)6-s + (−2.29 + 2.29i)7-s + (1.56 + 1.56i)8-s + (2.98 + 0.284i)9-s + 2.23i·10-s + (3.85 + 3.85i)11-s + (0.245 − 5.16i)12-s + (−0.766 + 3.52i)13-s − 7.24i·14-s + (1.28 − 1.16i)15-s + 1.04·16-s + 3.78·17-s + ⋯
L(s)  = 1  + (−1.11 + 1.11i)2-s + (0.998 + 0.0474i)3-s − 1.49i·4-s + (0.316 − 0.316i)5-s + (−1.16 + 1.06i)6-s + (−0.866 + 0.866i)7-s + (0.551 + 0.551i)8-s + (0.995 + 0.0948i)9-s + 0.706i·10-s + (1.16 + 1.16i)11-s + (0.0709 − 1.49i)12-s + (−0.212 + 0.977i)13-s − 1.93i·14-s + (0.330 − 0.300i)15-s + 0.261·16-s + 0.917·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.227 - 0.973i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.227 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.581693 + 0.733437i\)
\(L(\frac12)\) \(\approx\) \(0.581693 + 0.733437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0822i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.766 - 3.52i)T \)
good2 \( 1 + (1.57 - 1.57i)T - 2iT^{2} \)
7 \( 1 + (2.29 - 2.29i)T - 7iT^{2} \)
11 \( 1 + (-3.85 - 3.85i)T + 11iT^{2} \)
17 \( 1 - 3.78T + 17T^{2} \)
19 \( 1 + (1.28 + 1.28i)T + 19iT^{2} \)
23 \( 1 + 5.74T + 23T^{2} \)
29 \( 1 + 5.42iT - 29T^{2} \)
31 \( 1 + (3.95 + 3.95i)T + 31iT^{2} \)
37 \( 1 + (-6.99 + 6.99i)T - 37iT^{2} \)
41 \( 1 + (-0.872 + 0.872i)T - 41iT^{2} \)
43 \( 1 + 4.18iT - 43T^{2} \)
47 \( 1 + (1.56 + 1.56i)T + 47iT^{2} \)
53 \( 1 + 1.55iT - 53T^{2} \)
59 \( 1 + (4.00 + 4.00i)T + 59iT^{2} \)
61 \( 1 + 2.47T + 61T^{2} \)
67 \( 1 + (-5.37 - 5.37i)T + 67iT^{2} \)
71 \( 1 + (-1.46 + 1.46i)T - 71iT^{2} \)
73 \( 1 + (5.34 - 5.34i)T - 73iT^{2} \)
79 \( 1 + 4.01T + 79T^{2} \)
83 \( 1 + (-3.77 + 3.77i)T - 83iT^{2} \)
89 \( 1 + (-4.28 - 4.28i)T + 89iT^{2} \)
97 \( 1 + (12.3 + 12.3i)T + 97iT^{2} \)
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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

−12.81290072669855047968739846207, −12.00783320224780143784056976684, −9.906698347009022918407039067813, −9.497352040243518538898362715128, −8.959898951511368606509288826503, −7.81317354026124019291765480513, −6.86097395276446538449449738987, −5.93333989556668556158697735854, −4.08026538008411639579401248833, −2.03309061156349584107022664014, 1.23337265064511051837572764352, 3.03971400258354851316574527527, 3.65275178327517605268192696085, 6.23563665753497253014355036756, 7.58991240089770603124689300344, 8.482604707184296589092273812525, 9.496887968988775630391373656721, 10.09633559023765906555649378824, 10.85279274330046868219768731325, 12.14906872069009895288577200453

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