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Properties

Label 2-74-37.11-c5-0-10
Degree $2$
Conductor $74$
Sign $-0.255 + 0.966i$
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 + (7.85 − 13.6i)3-s + (7.99 − 13.8i)4-s + (−3.80 − 2.19i)5-s + 62.8i·6-s + (79.6 − 138. i)7-s + 63.9i·8-s + (−1.96 − 3.40i)9-s + 17.5·10-s + 161.·11-s + (−125. − 217. i)12-s + (−616. − 356. i)13-s + 637. i·14-s + (−59.8 + 34.5i)15-s + (−128 − 221. i)16-s + (−163. + 94.4i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.504 − 0.872i)3-s + (0.249 − 0.433i)4-s + (−0.0681 − 0.0393i)5-s + 0.712i·6-s + (0.614 − 1.06i)7-s + 0.353i·8-s + (−0.00807 − 0.0139i)9-s + 0.0556·10-s + 0.401·11-s + (−0.252 − 0.436i)12-s + (−1.01 − 0.584i)13-s + 0.869i·14-s + (−0.0686 + 0.0396i)15-s + (−0.125 − 0.216i)16-s + (−0.137 + 0.0792i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.255 + 0.966i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ -0.255 + 0.966i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.831808 - 1.07963i\)
\(L(\frac12)\) \(\approx\) \(0.831808 - 1.07963i\)
\(L(\frac{7}{2})\) 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 \( 1 + (7.32e3 - 3.95e3i)T \)
good3 \( 1 + (-7.85 + 13.6i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (3.80 + 2.19i)T + (1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-79.6 + 138. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 161.T + 1.61e5T^{2} \)
13 \( 1 + (616. + 356. i)T + (1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (163. - 94.4i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (684. + 395. i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + 1.76e3iT - 6.43e6T^{2} \)
29 \( 1 + 6.24e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.58e3iT - 2.86e7T^{2} \)
41 \( 1 + (671. - 1.16e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 - 1.87e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.96e3T + 2.29e8T^{2} \)
53 \( 1 + (-7.90e3 - 1.36e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-2.61e3 + 1.50e3i)T + (3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (4.95e3 + 2.85e3i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.73e4 + 3.00e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.41e4 + 2.45e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 - 4.19e4T + 2.07e9T^{2} \)
79 \( 1 + (-4.56e4 - 2.63e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-4.32e4 - 7.48e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (1.53e4 - 8.87e3i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 3.59e4iT - 8.58e9T^{2} \)
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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

−13.48272817277217080355244088309, −12.24839915975077878477150018803, −10.87022164010894127364568640117, −9.805678571851759115798540325881, −8.206491946291546388493793616932, −7.63350342576960715536977019197, −6.52436812256940709976508467561, −4.54756472950554036104895929464, −2.22593143829808133171325861233, −0.68767450260201961688168529280, 1.97194076966496057459233955274, 3.57975517174065051505253804470, 5.12919469990658133928620034252, 7.09928371363320717592532172959, 8.680174864529717335285356933021, 9.236730034546155016980427928441, 10.35677390152734403164838815454, 11.63343755584895972079022748978, 12.44194792779778132494683464082, 14.24015033134884840955690638424

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