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Properties

Label 2-195-3.2-c2-0-15
Degree $2$
Conductor $195$
Sign $-0.183 + 0.983i$
Analytic cond. $5.31336$
Root an. cond. $2.30507$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.93i·2-s + (−2.94 − 0.549i)3-s + 0.267·4-s + 2.23i·5-s + (−1.06 + 5.69i)6-s + 10.3·7-s − 8.24i·8-s + (8.39 + 3.24i)9-s + 4.32·10-s − 2.40i·11-s + (−0.788 − 0.146i)12-s − 3.60·13-s − 19.9i·14-s + (1.22 − 6.59i)15-s − 14.8·16-s − 19.9i·17-s + ⋯
L(s)  = 1  − 0.966i·2-s + (−0.983 − 0.183i)3-s + 0.0668·4-s + 0.447i·5-s + (−0.176 + 0.949i)6-s + 1.47·7-s − 1.03i·8-s + (0.932 + 0.360i)9-s + 0.432·10-s − 0.218i·11-s + (−0.0656 − 0.0122i)12-s − 0.277·13-s − 1.42i·14-s + (0.0818 − 0.439i)15-s − 0.928·16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.183 + 0.983i$
Analytic conductor: \(5.31336\)
Root analytic conductor: \(2.30507\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1),\ -0.183 + 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.912755 - 1.09844i\)
\(L(\frac12)\) \(\approx\) \(0.912755 - 1.09844i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.94 + 0.549i)T \)
5 \( 1 - 2.23iT \)
13 \( 1 + 3.60T \)
good2 \( 1 + 1.93iT - 4T^{2} \)
7 \( 1 - 10.3T + 49T^{2} \)
11 \( 1 + 2.40iT - 121T^{2} \)
17 \( 1 + 19.9iT - 289T^{2} \)
19 \( 1 - 13.5T + 361T^{2} \)
23 \( 1 + 22.7iT - 529T^{2} \)
29 \( 1 - 26.6iT - 841T^{2} \)
31 \( 1 + 10.9T + 961T^{2} \)
37 \( 1 - 44.4T + 1.36e3T^{2} \)
41 \( 1 + 46.7iT - 1.68e3T^{2} \)
43 \( 1 - 51.3T + 1.84e3T^{2} \)
47 \( 1 - 27.8iT - 2.20e3T^{2} \)
53 \( 1 + 5.34iT - 2.80e3T^{2} \)
59 \( 1 - 62.4iT - 3.48e3T^{2} \)
61 \( 1 + 71.6T + 3.72e3T^{2} \)
67 \( 1 - 16.7T + 4.48e3T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 + 11.0T + 5.32e3T^{2} \)
79 \( 1 - 76.8T + 6.24e3T^{2} \)
83 \( 1 - 58.6iT - 6.88e3T^{2} \)
89 \( 1 + 2.00iT - 7.92e3T^{2} \)
97 \( 1 + 169.T + 9.40e3T^{2} \)
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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

−11.79920618281966238490320456478, −11.06766264431691922848286508679, −10.60889855682942552428989009912, −9.388734431620740851553088713895, −7.70105417658644699394408344388, −6.88794710938101845414171250350, −5.46786710893817483241455282410, −4.32937518573373766898828697090, −2.51588565620950687746753514803, −1.07015488368625122595447103091, 1.63523772313594148139790454175, 4.40699630083484525463012652115, 5.31133175737080781452306929185, 6.13469252326641443047850672950, 7.49207755016612536287471571310, 8.095443978073058475569193461986, 9.521457001455423005973923843947, 10.88261585218500138122043351403, 11.48493650947673856850898906306, 12.33905713709040029238025750911

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