Properties

Label 2-74-37.10-c5-0-14
Degree $2$
Conductor $74$
Sign $-0.354 - 0.935i$
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  + (2 − 3.46i)2-s + (−12.1 − 21.1i)3-s + (−7.99 − 13.8i)4-s + (−5.21 − 9.03i)5-s − 97.4·6-s + (−70.8 − 122. i)7-s − 63.9·8-s + (−175. + 303. i)9-s − 41.7·10-s + 313.·11-s + (−194. + 337. i)12-s + (−152. − 264. i)13-s − 566.·14-s + (−127. + 220. i)15-s + (−128 + 221. i)16-s + (26.6 − 46.1i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.781 − 1.35i)3-s + (−0.249 − 0.433i)4-s + (−0.0933 − 0.161i)5-s − 1.10·6-s + (−0.546 − 0.946i)7-s − 0.353·8-s + (−0.722 + 1.25i)9-s − 0.132·10-s + 0.780·11-s + (−0.390 + 0.676i)12-s + (−0.250 − 0.433i)13-s − 0.773·14-s + (−0.145 + 0.252i)15-s + (−0.125 + 0.216i)16-s + (0.0223 − 0.0387i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.354 - 0.935i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ -0.354 - 0.935i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.473111 + 0.685303i\)
\(L(\frac12)\) \(\approx\) \(0.473111 + 0.685303i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 3.46i)T \)
37 \( 1 + (1.77e3 + 8.13e3i)T \)
good3 \( 1 + (12.1 + 21.1i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (5.21 + 9.03i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (70.8 + 122. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 313.T + 1.61e5T^{2} \)
13 \( 1 + (152. + 264. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-26.6 + 46.1i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.48e3 - 2.57e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + 253.T + 6.43e6T^{2} \)
29 \( 1 + 7.63e3T + 2.05e7T^{2} \)
31 \( 1 + 145.T + 2.86e7T^{2} \)
41 \( 1 + (1.87e3 + 3.24e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + 5.79e3T + 1.47e8T^{2} \)
47 \( 1 + 1.05e4T + 2.29e8T^{2} \)
53 \( 1 + (5.14e3 - 8.90e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.21e4 + 2.09e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.58e4 + 2.74e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (8.62e3 + 1.49e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-1.42e4 - 2.47e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + 2.38e4T + 2.07e9T^{2} \)
79 \( 1 + (-3.78e4 - 6.55e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-5.28e4 + 9.14e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (4.88e4 - 8.46e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 6.33e4T + 8.58e9T^{2} \)
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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

−12.63729116451178127641237888831, −12.02557927392527650430708693521, −10.91120617636667602120268918053, −9.705515599351316001481450037743, −7.81636087595414211008184771868, −6.72474853542709456287617424341, −5.57514501466426342543285984828, −3.70674660925172302026386565523, −1.58104541565713231708480484280, −0.36854555964785389397430756848, 3.31492053634552597879855681789, 4.72950847646033918530054585125, 5.72484583828667285757498129498, 6.95310913309072018191807135194, 9.029470238135622783471661569892, 9.610643261173917890700546249611, 11.21436115232799434774998754835, 11.91916723418955486497259143180, 13.34490258336837423233923700759, 14.87798082150457885747406466437

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