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Properties

Label 2-74-37.36-c3-0-5
Degree $2$
Conductor $74$
Sign $0.951 + 0.307i$
Analytic cond. $4.36614$
Root an. cond. $2.08953$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 7.35·3-s − 4·4-s + 16.2i·5-s − 14.7i·6-s + 31.9·7-s + 8i·8-s + 27.0·9-s + 32.4·10-s − 58.7·11-s − 29.4·12-s − 35.5i·13-s − 63.8i·14-s + 119. i·15-s + 16·16-s − 109. i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.41·3-s − 0.5·4-s + 1.45i·5-s − 1.00i·6-s + 1.72·7-s + 0.353i·8-s + 1.00·9-s + 1.02·10-s − 1.60·11-s − 0.707·12-s − 0.759i·13-s − 1.21i·14-s + 2.05i·15-s + 0.250·16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.951 + 0.307i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ 0.951 + 0.307i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.21790 - 0.349705i\)
\(L(\frac12)\) \(\approx\) \(2.21790 - 0.349705i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
37 \( 1 + (69.2 - 214. i)T \)
good3 \( 1 - 7.35T + 27T^{2} \)
5 \( 1 - 16.2iT - 125T^{2} \)
7 \( 1 - 31.9T + 343T^{2} \)
11 \( 1 + 58.7T + 1.33e3T^{2} \)
13 \( 1 + 35.5iT - 2.19e3T^{2} \)
17 \( 1 + 109. iT - 4.91e3T^{2} \)
19 \( 1 + 2.05iT - 6.85e3T^{2} \)
23 \( 1 - 86.8iT - 1.21e4T^{2} \)
29 \( 1 - 50.6iT - 2.43e4T^{2} \)
31 \( 1 + 252. iT - 2.97e4T^{2} \)
41 \( 1 - 9.98T + 6.89e4T^{2} \)
43 \( 1 - 300. iT - 7.95e4T^{2} \)
47 \( 1 + 118.T + 1.03e5T^{2} \)
53 \( 1 + 250.T + 1.48e5T^{2} \)
59 \( 1 + 122. iT - 2.05e5T^{2} \)
61 \( 1 + 560. iT - 2.26e5T^{2} \)
67 \( 1 + 214.T + 3.00e5T^{2} \)
71 \( 1 + 147.T + 3.57e5T^{2} \)
73 \( 1 - 478.T + 3.89e5T^{2} \)
79 \( 1 - 18.4iT - 4.93e5T^{2} \)
83 \( 1 - 1.18e3T + 5.71e5T^{2} \)
89 \( 1 - 558. iT - 7.04e5T^{2} \)
97 \( 1 - 1.29e3iT - 9.12e5T^{2} \)
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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

−14.02139405151469287068544321810, −13.32365829782270324963363639069, −11.49403230259608028192841112590, −10.74663910784562826346980661304, −9.631791272395210827241121978557, −8.021605218445947063641431735432, −7.63515650736370921877598296653, −5.05156997286459255184733962116, −3.13727408686809356813724977389, −2.29930446899146428342066576605, 1.83431965271329396847907830165, 4.30461050133311448316020976486, 5.29942635609025221030273285925, 7.69634147902303215504698516547, 8.406549577542145318452588000843, 8.847673400421339524467368201673, 10.54565806368592595540314797647, 12.38274125461648763191828325249, 13.34388611816610238689777862971, 14.25165505333321238312473190680

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