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Properties

Label 2-74-37.26-c3-0-0
Degree $2$
Conductor $74$
Sign $0.251 - 0.967i$
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  + (−1 − 1.73i)2-s + (1.63 − 2.83i)3-s + (−1.99 + 3.46i)4-s + (−9.76 + 16.9i)5-s − 6.55·6-s + (−7.16 + 12.4i)7-s + 7.99·8-s + (8.12 + 14.0i)9-s + 39.0·10-s − 40.4·11-s + (6.55 + 11.3i)12-s + (29.3 − 50.8i)13-s + 28.6·14-s + (32.0 + 55.4i)15-s + (−8 − 13.8i)16-s + (43.7 + 75.7i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.315 − 0.546i)3-s + (−0.249 + 0.433i)4-s + (−0.873 + 1.51i)5-s − 0.445·6-s + (−0.387 + 0.670i)7-s + 0.353·8-s + (0.301 + 0.521i)9-s + 1.23·10-s − 1.10·11-s + (0.157 + 0.273i)12-s + (0.626 − 1.08i)13-s + 0.547·14-s + (0.550 + 0.954i)15-s + (−0.125 − 0.216i)16-s + (0.623 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.251 - 0.967i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ 0.251 - 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.623696 + 0.482464i\)
\(L(\frac12)\) \(\approx\) \(0.623696 + 0.482464i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
37 \( 1 + (-24.0 + 223. i)T \)
good3 \( 1 + (-1.63 + 2.83i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (9.76 - 16.9i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (7.16 - 12.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + 40.4T + 1.33e3T^{2} \)
13 \( 1 + (-29.3 + 50.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-43.7 - 75.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (55.4 - 96.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 124.T + 1.21e4T^{2} \)
29 \( 1 - 169.T + 2.43e4T^{2} \)
31 \( 1 + 121.T + 2.97e4T^{2} \)
41 \( 1 + (120. - 208. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 - 222.T + 7.95e4T^{2} \)
47 \( 1 + 242.T + 1.03e5T^{2} \)
53 \( 1 + (-255. - 441. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-138. - 239. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-185. + 321. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-55.8 + 96.7i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-304. + 526. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 64.7T + 3.89e5T^{2} \)
79 \( 1 + (-444. + 769. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-643. - 1.11e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (201. + 349. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.82e3T + 9.12e5T^{2} \)
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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

−14.26664437741719602921346080222, −12.94162557038199320944486073926, −12.18705685217715712232531622234, −10.62242463050721607594695005900, −10.38527503243495328043913607977, −8.146260579594997999749710444152, −7.75535024864781510885634392034, −6.08843298229702989755442773077, −3.58192854373335035608437158625, −2.40452090431143133372207346355, 0.54919644440612391123735505458, 3.97987066633208370631233787498, 4.97710166721376665152160135037, 6.92659877086560009307799482759, 8.231727605184260814100920468814, 9.087868742282336266149052534378, 10.10773098882946321612775163440, 11.68115252038965650482419688675, 12.90661753758385585420009836299, 13.88085546188074493393247491680

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