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Properties

Label 2-195-65.18-c1-0-4
Degree $2$
Conductor $195$
Sign $0.395 - 0.918i$
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  + 0.470i·2-s + (0.707 + 0.707i)3-s + 1.77·4-s + (0.0206 + 2.23i)5-s + (−0.332 + 0.332i)6-s − 1.17·7-s + 1.77i·8-s + 1.00i·9-s + (−1.05 + 0.00971i)10-s + (−4.60 − 4.60i)11-s + (1.25 + 1.25i)12-s + (2.91 − 2.12i)13-s − 0.555i·14-s + (−1.56 + 1.59i)15-s + 2.72·16-s + (3.73 + 3.73i)17-s + ⋯
L(s)  = 1  + 0.332i·2-s + (0.408 + 0.408i)3-s + 0.889·4-s + (0.00923 + 0.999i)5-s + (−0.135 + 0.135i)6-s − 0.445·7-s + 0.628i·8-s + 0.333i·9-s + (−0.332 + 0.00307i)10-s + (−1.38 − 1.38i)11-s + (0.363 + 0.363i)12-s + (0.808 − 0.588i)13-s − 0.148i·14-s + (−0.404 + 0.412i)15-s + 0.680·16-s + (0.906 + 0.906i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.395 - 0.918i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25435 + 0.825741i\)
\(L(\frac12)\) \(\approx\) \(1.25435 + 0.825741i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0206 - 2.23i)T \)
13 \( 1 + (-2.91 + 2.12i)T \)
good2 \( 1 - 0.470iT - 2T^{2} \)
7 \( 1 + 1.17T + 7T^{2} \)
11 \( 1 + (4.60 + 4.60i)T + 11iT^{2} \)
17 \( 1 + (-3.73 - 3.73i)T + 17iT^{2} \)
19 \( 1 + (2.02 + 2.02i)T + 19iT^{2} \)
23 \( 1 + (0.0111 - 0.0111i)T - 23iT^{2} \)
29 \( 1 + 0.795iT - 29T^{2} \)
31 \( 1 + (-5.37 + 5.37i)T - 31iT^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 + (-5.70 + 5.70i)T - 41iT^{2} \)
43 \( 1 + (0.401 - 0.401i)T - 43iT^{2} \)
47 \( 1 + 6.70T + 47T^{2} \)
53 \( 1 + (4.45 + 4.45i)T + 53iT^{2} \)
59 \( 1 + (0.197 - 0.197i)T - 59iT^{2} \)
61 \( 1 + 7.89T + 61T^{2} \)
67 \( 1 - 7.46iT - 67T^{2} \)
71 \( 1 + (1.19 - 1.19i)T - 71iT^{2} \)
73 \( 1 + 5.10iT - 73T^{2} \)
79 \( 1 - 7.80iT - 79T^{2} \)
83 \( 1 + 4.00T + 83T^{2} \)
89 \( 1 + (9.19 - 9.19i)T - 89iT^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−12.86513325012111953480868057392, −11.31703056435616939975766491028, −10.74166153124301332318096993292, −10.04172155271456740333079648610, −8.315644373250711791341315088984, −7.76413583274508868916286863271, −6.33031995324499031225205343742, −5.67588705316033418378318223979, −3.45646753477830115906886361658, −2.67046574412618899088362274775, 1.62358452331468809678339195602, 3.01216833448697350380289457708, 4.69640753511998867366230899173, 6.13802317734326061664384497210, 7.33828799039543663214795634930, 8.152786060549238764079525891449, 9.513993007217301682329222854891, 10.25022132954588394278085781192, 11.57179038183697167627478296219, 12.53929336286185605447009110255

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