Properties

Label 2-74-37.14-c4-0-3
Degree $2$
Conductor $74$
Sign $0.997 - 0.0769i$
Analytic cond. $7.64937$
Root an. cond. $2.76575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.73 + 0.732i)2-s + (−9.68 + 5.59i)3-s + (6.92 − 4i)4-s + (−24.8 − 6.64i)5-s + (22.3 − 22.3i)6-s + (−19.9 − 34.4i)7-s + (−15.9 + 16i)8-s + (22.0 − 38.1i)9-s + 72.6·10-s + 64.8i·11-s + (−44.7 + 77.4i)12-s + (111. + 29.9i)13-s + (79.6 + 79.6i)14-s + (277. − 74.3i)15-s + (31.9 − 55.4i)16-s + (−15.2 − 57.0i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−1.07 + 0.621i)3-s + (0.433 − 0.250i)4-s + (−0.992 − 0.265i)5-s + (0.621 − 0.621i)6-s + (−0.406 − 0.703i)7-s + (−0.249 + 0.250i)8-s + (0.271 − 0.470i)9-s + 0.726·10-s + 0.536i·11-s + (−0.310 + 0.537i)12-s + (0.661 + 0.177i)13-s + (0.406 + 0.406i)14-s + (1.23 − 0.330i)15-s + (0.124 − 0.216i)16-s + (−0.0529 − 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0769i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.997 - 0.0769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.997 - 0.0769i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ 0.997 - 0.0769i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.547912 + 0.0211177i\)
\(L(\frac12)\) \(\approx\) \(0.547912 + 0.0211177i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.73 - 0.732i)T \)
37 \( 1 + (17.1 + 1.36e3i)T \)
good3 \( 1 + (9.68 - 5.59i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (24.8 + 6.64i)T + (541. + 312.5i)T^{2} \)
7 \( 1 + (19.9 + 34.4i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 - 64.8iT - 1.46e4T^{2} \)
13 \( 1 + (-111. - 29.9i)T + (2.47e4 + 1.42e4i)T^{2} \)
17 \( 1 + (15.2 + 57.0i)T + (-7.23e4 + 4.17e4i)T^{2} \)
19 \( 1 + (-628. - 168. i)T + (1.12e5 + 6.51e4i)T^{2} \)
23 \( 1 + (-152. + 152. i)T - 2.79e5iT^{2} \)
29 \( 1 + (-312. - 312. i)T + 7.07e5iT^{2} \)
31 \( 1 + (68.3 + 68.3i)T + 9.23e5iT^{2} \)
41 \( 1 + (1.70e3 - 984. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-908. + 908. i)T - 3.41e6iT^{2} \)
47 \( 1 - 2.23e3T + 4.87e6T^{2} \)
53 \( 1 + (22.0 - 38.2i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (1.07e3 + 4.01e3i)T + (-1.04e7 + 6.05e6i)T^{2} \)
61 \( 1 + (307. - 1.14e3i)T + (-1.19e7 - 6.92e6i)T^{2} \)
67 \( 1 + (-6.82e3 + 3.93e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (-4.03e3 - 6.99e3i)T + (-1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 - 1.31e3iT - 2.83e7T^{2} \)
79 \( 1 + (-9.16e3 - 2.45e3i)T + (3.37e7 + 1.94e7i)T^{2} \)
83 \( 1 + (-1.30e3 + 2.26e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + (-4.27e3 + 1.14e3i)T + (5.43e7 - 3.13e7i)T^{2} \)
97 \( 1 + (5.67e3 - 5.67e3i)T - 8.85e7iT^{2} \)
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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

−13.94067335166264238114765283982, −12.29163446962157608287795616502, −11.43214834523673715006564233794, −10.53164737173966025548388354602, −9.505633345579998811267199036504, −7.964876545149829091586132282791, −6.82397079346257415786125445069, −5.28628943380072563076982000666, −3.86036041347009890465213802059, −0.64210389279915573976668283857, 0.834632805836137244273669197466, 3.26785504150180704454644135416, 5.60007335857681757360032348193, 6.76846351160832924677260205085, 7.912388500379089066506086351727, 9.224502722313432339600411940639, 10.81251456972377573607336010654, 11.66041073829049732564307211154, 12.17800671696888025915577817769, 13.52205163811933571229122587872

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