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Properties

Label 2-195-39.8-c1-0-13
Degree $2$
Conductor $195$
Sign $0.936 + 0.350i$
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  + (0.483 − 0.483i)2-s + (1.53 − 0.793i)3-s + 1.53i·4-s + (0.707 − 0.707i)5-s + (0.361 − 1.12i)6-s + (−1.24 + 1.24i)7-s + (1.70 + 1.70i)8-s + (1.74 − 2.44i)9-s − 0.684i·10-s + (0.387 + 0.387i)11-s + (1.21 + 2.35i)12-s + (−0.874 − 3.49i)13-s + 1.20i·14-s + (0.527 − 1.64i)15-s − 1.41·16-s − 6.75·17-s + ⋯
L(s)  = 1  + (0.342 − 0.342i)2-s + (0.888 − 0.458i)3-s + 0.765i·4-s + (0.316 − 0.316i)5-s + (0.147 − 0.460i)6-s + (−0.470 + 0.470i)7-s + (0.604 + 0.604i)8-s + (0.580 − 0.814i)9-s − 0.216i·10-s + (0.116 + 0.116i)11-s + (0.350 + 0.680i)12-s + (−0.242 − 0.970i)13-s + 0.322i·14-s + (0.136 − 0.425i)15-s − 0.352·16-s − 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.936 + 0.350i$
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: \((2,\ 195,\ (\ :1/2),\ 0.936 + 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76639 - 0.320048i\)
\(L(\frac12)\) \(\approx\) \(1.76639 - 0.320048i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53 + 0.793i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.874 + 3.49i)T \)
good2 \( 1 + (-0.483 + 0.483i)T - 2iT^{2} \)
7 \( 1 + (1.24 - 1.24i)T - 7iT^{2} \)
11 \( 1 + (-0.387 - 0.387i)T + 11iT^{2} \)
17 \( 1 + 6.75T + 17T^{2} \)
19 \( 1 + (-1.33 - 1.33i)T + 19iT^{2} \)
23 \( 1 + 1.69T + 23T^{2} \)
29 \( 1 + 3.85iT - 29T^{2} \)
31 \( 1 + (-4.06 - 4.06i)T + 31iT^{2} \)
37 \( 1 + (2.36 - 2.36i)T - 37iT^{2} \)
41 \( 1 + (5.72 - 5.72i)T - 41iT^{2} \)
43 \( 1 + 11.7iT - 43T^{2} \)
47 \( 1 + (-5.99 - 5.99i)T + 47iT^{2} \)
53 \( 1 - 8.81iT - 53T^{2} \)
59 \( 1 + (1.30 + 1.30i)T + 59iT^{2} \)
61 \( 1 + 1.16T + 61T^{2} \)
67 \( 1 + (4.66 + 4.66i)T + 67iT^{2} \)
71 \( 1 + (-0.915 + 0.915i)T - 71iT^{2} \)
73 \( 1 + (-8.11 + 8.11i)T - 73iT^{2} \)
79 \( 1 + 3.85T + 79T^{2} \)
83 \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \)
89 \( 1 + (-5.99 - 5.99i)T + 89iT^{2} \)
97 \( 1 + (-1.03 - 1.03i)T + 97iT^{2} \)
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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

−12.50599128413275724846523436246, −11.97022972618033940577261585280, −10.47927394818871541773175381256, −9.236729748686986813244244338058, −8.480714032250023074732397883041, −7.51684430150003795746747501892, −6.29849469997334618277215443886, −4.61401958646306362769892000225, −3.26261795495965523380883592765, −2.21427252696778100846015715447, 2.15428021598666875713182826872, 3.89755686382527450310051753022, 4.91776232054509304681689325308, 6.47144949495955844160593336869, 7.17376307003843530641876801109, 8.787779707340810311002890395475, 9.623051551345906570298891091936, 10.38767715513051820042553835338, 11.35457632695614949296265616307, 13.12197399703401249508752689705

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