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Properties

Label 2-82-41.37-c1-0-3
Degree $2$
Conductor $82$
Sign $0.767 + 0.640i$
Analytic cond. $0.654773$
Root an. cond. $0.809180$
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  + (0.809 − 0.587i)2-s − 0.618·3-s + (0.309 − 0.951i)4-s + (0.5 − 1.53i)5-s + (−0.500 + 0.363i)6-s + (2.11 + 1.53i)7-s + (−0.309 − 0.951i)8-s − 2.61·9-s + (−0.5 − 1.53i)10-s + (0.927 + 2.85i)11-s + (−0.190 + 0.587i)12-s + (−5.04 + 3.66i)13-s + 2.61·14-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (−1.30 − 4.02i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s − 0.356·3-s + (0.154 − 0.475i)4-s + (0.223 − 0.688i)5-s + (−0.204 + 0.148i)6-s + (0.800 + 0.581i)7-s + (−0.109 − 0.336i)8-s − 0.872·9-s + (−0.158 − 0.486i)10-s + (0.279 + 0.860i)11-s + (−0.0551 + 0.169i)12-s + (−1.39 + 1.01i)13-s + 0.699·14-s + (−0.0797 + 0.245i)15-s + (−0.202 − 0.146i)16-s + (−0.317 − 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(82\)    =    \(2 \cdot 41\)
Sign: $0.767 + 0.640i$
Analytic conductor: \(0.654773\)
Root analytic conductor: \(0.809180\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{82} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 82,\ (\ :1/2),\ 0.767 + 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09218 - 0.395679i\)
\(L(\frac12)\) \(\approx\) \(1.09218 - 0.395679i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-2.19 + 6.01i)T \)
good3 \( 1 + 0.618T + 3T^{2} \)
5 \( 1 + (-0.5 + 1.53i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.11 - 1.53i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.927 - 2.85i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (5.04 - 3.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.30 + 4.02i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.42 - 1.03i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.30 + 0.951i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.163 - 0.502i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.88 + 5.79i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.454 + 1.40i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (7.85 - 5.70i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-1.23 + 0.898i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.66 + 11.2i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.11 + 2.99i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.80 - 2.04i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (4.76 - 14.6i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.19 - 3.66i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 16.1T + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 + (-5.04 - 3.66i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.69 + 5.20i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.35919543657035728402401040983, −12.98022314929770790114314171643, −11.84534463284178099796781617039, −11.52629822953851799264685749018, −9.789639595161949665075009716120, −8.823991793887070615223546868502, −7.08965134159017573125739836929, −5.39497494461085526464031344184, −4.66514199240754001238760529303, −2.23702309368809664791477419693, 3.05161853005708542495495633742, 4.92820331480323583749960170799, 6.08972561009852941023174852754, 7.38023786300592133352439022850, 8.541405602564407266591647301522, 10.42711629890546086583448115419, 11.20353784443618874093963915424, 12.32644670808381425022027492419, 13.67226503227399745044615410632, 14.48943712287325073406255976045

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