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Properties

Label 2-74-37.27-c5-0-6
Degree $2$
Conductor $74$
Sign $0.0571 + 0.998i$
Analytic cond. $11.8684$
Root an. cond. $3.44505$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.46 − 2i)2-s + (−1.00 − 1.73i)3-s + (7.99 + 13.8i)4-s + (−82.7 + 47.7i)5-s + 8.01i·6-s + (45.7 + 79.2i)7-s − 63.9i·8-s + (119. − 206. i)9-s + 382.·10-s − 180.·11-s + (16.0 − 27.7i)12-s + (337. − 194. i)13-s − 365. i·14-s + (165. + 95.7i)15-s + (−128 + 221. i)16-s + (−358. − 207. i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.0642 − 0.111i)3-s + (0.249 + 0.433i)4-s + (−1.48 + 0.854i)5-s + 0.0909i·6-s + (0.352 + 0.610i)7-s − 0.353i·8-s + (0.491 − 0.851i)9-s + 1.20·10-s − 0.449·11-s + (0.0321 − 0.0556i)12-s + (0.554 − 0.319i)13-s − 0.498i·14-s + (0.190 + 0.109i)15-s + (−0.125 + 0.216i)16-s + (−0.301 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0571 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0571 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.0571 + 0.998i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ 0.0571 + 0.998i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.526642 - 0.497336i\)
\(L(\frac12)\) \(\approx\) \(0.526642 - 0.497336i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.46 + 2i)T \)
37 \( 1 + (-6.39e3 - 5.33e3i)T \)
good3 \( 1 + (1.00 + 1.73i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (82.7 - 47.7i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-45.7 - 79.2i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 180.T + 1.61e5T^{2} \)
13 \( 1 + (-337. + 194. i)T + (1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (358. + 207. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-530. + 306. i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + 3.52e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.14e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.48e3iT - 2.86e7T^{2} \)
41 \( 1 + (-6.29e3 - 1.09e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + 1.93e3iT - 1.47e8T^{2} \)
47 \( 1 + 7.29e3T + 2.29e8T^{2} \)
53 \( 1 + (-1.14e4 + 1.97e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-8.22e3 - 4.75e3i)T + (3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (3.38e3 - 1.95e3i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.76e4 - 3.05e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (4.83e3 + 8.37e3i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + 7.15e4T + 2.07e9T^{2} \)
79 \( 1 + (-7.53e4 + 4.34e4i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (2.48e3 - 4.30e3i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (6.89e4 + 3.98e4i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 8.05e4iT - 8.58e9T^{2} \)
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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

−12.98904143283231718286874904904, −11.83766879372140983976341933012, −11.28889890658401391246963239378, −10.07503695928052824856062152788, −8.565164611976027715624914214916, −7.65519674695809265170980770409, −6.46121514695924185652427952979, −4.16399758293850973370237374365, −2.78545177871265502749832290581, −0.46274027602484711004524164029, 1.21647238008156824755010912776, 3.98568825153838748118810381567, 5.17481003044298667866242360806, 7.31706398255111617851119243951, 7.927573502764898999404774097477, 9.047631168681921493481913774216, 10.62641198745003519543308203414, 11.39738701343260555768493223826, 12.69375750027118982425191860381, 13.88937894888839225667791698992

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