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Properties

Label 2-195-39.5-c1-0-14
Degree $2$
Conductor $195$
Sign $0.994 + 0.103i$
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  + (1.11 + 1.11i)2-s + (0.455 − 1.67i)3-s + 0.503i·4-s + (−0.707 − 0.707i)5-s + (2.37 − 1.36i)6-s + (1.46 + 1.46i)7-s + (1.67 − 1.67i)8-s + (−2.58 − 1.52i)9-s − 1.58i·10-s + (−0.292 + 0.292i)11-s + (0.841 + 0.229i)12-s + (2.19 + 2.86i)13-s + 3.28i·14-s + (−1.50 + 0.859i)15-s + 4.75·16-s − 2.78·17-s + ⋯
L(s)  = 1  + (0.791 + 0.791i)2-s + (0.262 − 0.964i)3-s + 0.251i·4-s + (−0.316 − 0.316i)5-s + (0.971 − 0.555i)6-s + (0.554 + 0.554i)7-s + (0.591 − 0.591i)8-s + (−0.861 − 0.507i)9-s − 0.500i·10-s + (−0.0883 + 0.0883i)11-s + (0.242 + 0.0661i)12-s + (0.608 + 0.793i)13-s + 0.876i·14-s + (−0.388 + 0.221i)15-s + 1.18·16-s − 0.674·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.994 + 0.103i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84044 - 0.0956351i\)
\(L(\frac12)\) \(\approx\) \(1.84044 - 0.0956351i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.455 + 1.67i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-2.19 - 2.86i)T \)
good2 \( 1 + (-1.11 - 1.11i)T + 2iT^{2} \)
7 \( 1 + (-1.46 - 1.46i)T + 7iT^{2} \)
11 \( 1 + (0.292 - 0.292i)T - 11iT^{2} \)
17 \( 1 + 2.78T + 17T^{2} \)
19 \( 1 + (1.21 - 1.21i)T - 19iT^{2} \)
23 \( 1 + 5.66T + 23T^{2} \)
29 \( 1 - 7.34iT - 29T^{2} \)
31 \( 1 + (1.98 - 1.98i)T - 31iT^{2} \)
37 \( 1 + (3.02 + 3.02i)T + 37iT^{2} \)
41 \( 1 + (-3.08 - 3.08i)T + 41iT^{2} \)
43 \( 1 - 0.831iT - 43T^{2} \)
47 \( 1 + (-8.76 + 8.76i)T - 47iT^{2} \)
53 \( 1 - 0.258iT - 53T^{2} \)
59 \( 1 + (2.38 - 2.38i)T - 59iT^{2} \)
61 \( 1 + 3.66T + 61T^{2} \)
67 \( 1 + (-9.29 + 9.29i)T - 67iT^{2} \)
71 \( 1 + (-7.88 - 7.88i)T + 71iT^{2} \)
73 \( 1 + (11.5 + 11.5i)T + 73iT^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 + (4.97 - 4.97i)T - 89iT^{2} \)
97 \( 1 + (8.41 - 8.41i)T - 97iT^{2} \)
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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

−12.66080695265832179122877635699, −11.92564011180330881608102575638, −10.78694455360575079127738802396, −9.089634476396833869076689264642, −8.227793685794382095390922045543, −7.16375828526980492300220574851, −6.25498763889650057906814241753, −5.23086545017406188587400885264, −3.89963085013348978113903934334, −1.80031461742363514737154776835, 2.51096536283822442897808805784, 3.80998367616223847041837639525, 4.46113621801994623143053603351, 5.78729912352326129251611666067, 7.69857107299788812027064857348, 8.468770665312453918670338141352, 9.970414480131109065006938262110, 10.92240219721066076886093563095, 11.26448243697573374178714462413, 12.47158039694902370121716271729

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