Properties

Label 2-74-37.36-c3-0-8
Degree $2$
Conductor $74$
Sign $0.213 + 0.976i$
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 + 9.84·3-s − 4·4-s − 14.4i·5-s − 19.6i·6-s − 22.1·7-s + 8i·8-s + 69.8·9-s − 28.8·10-s + 21.0·11-s − 39.3·12-s + 77.2i·13-s + 44.3i·14-s − 141. i·15-s + 16·16-s − 73.4i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.89·3-s − 0.5·4-s − 1.28i·5-s − 1.33i·6-s − 1.19·7-s + 0.353i·8-s + 2.58·9-s − 0.911·10-s + 0.576·11-s − 0.947·12-s + 1.64i·13-s + 0.847i·14-s − 2.44i·15-s + 0.250·16-s − 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.213 + 0.976i$
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: \((2,\ 74,\ (\ :3/2),\ 0.213 + 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.78996 - 1.44096i\)
\(L(\frac12)\) \(\approx\) \(1.78996 - 1.44096i\)
\(L(\frac{5}{2})\) 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 \( 1 + (219. - 48.0i)T \)
good3 \( 1 - 9.84T + 27T^{2} \)
5 \( 1 + 14.4iT - 125T^{2} \)
7 \( 1 + 22.1T + 343T^{2} \)
11 \( 1 - 21.0T + 1.33e3T^{2} \)
13 \( 1 - 77.2iT - 2.19e3T^{2} \)
17 \( 1 + 73.4iT - 4.91e3T^{2} \)
19 \( 1 - 68.5iT - 6.85e3T^{2} \)
23 \( 1 + 25.3iT - 1.21e4T^{2} \)
29 \( 1 - 268. iT - 2.43e4T^{2} \)
31 \( 1 + 86.5iT - 2.97e4T^{2} \)
41 \( 1 - 5.19T + 6.89e4T^{2} \)
43 \( 1 + 125. iT - 7.95e4T^{2} \)
47 \( 1 + 181.T + 1.03e5T^{2} \)
53 \( 1 - 63.1T + 1.48e5T^{2} \)
59 \( 1 + 207. iT - 2.05e5T^{2} \)
61 \( 1 - 299. iT - 2.26e5T^{2} \)
67 \( 1 + 142.T + 3.00e5T^{2} \)
71 \( 1 + 501.T + 3.57e5T^{2} \)
73 \( 1 - 200.T + 3.89e5T^{2} \)
79 \( 1 + 1.07e3iT - 4.93e5T^{2} \)
83 \( 1 + 1.12e3T + 5.71e5T^{2} \)
89 \( 1 - 154. iT - 7.04e5T^{2} \)
97 \( 1 + 829. iT - 9.12e5T^{2} \)
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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

−13.74495382750959728075767661838, −12.88014646361237273160923861618, −12.06489457474869742211352130316, −9.903781535194843473758848170952, −9.149739208433809696435909572424, −8.694414655761520499350492428262, −7.03076724246144656422681832384, −4.46526820581421215375895934330, −3.33445681914912380303200567535, −1.65766942906267841588348876238, 2.84243861556447914107317339178, 3.70312467222885868317765993402, 6.37773379501088365090330668500, 7.37052451512932012957728949097, 8.415606221179064082673947934740, 9.634689055225641360145818752463, 10.39776607205581100140950743979, 12.78962938711871678284063575000, 13.50678106764594623689289253135, 14.50224484772684318414084524417

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