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Properties

Label 2-74-37.11-c1-0-3
Degree $2$
Conductor $74$
Sign $0.783 + 0.621i$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.366 − 0.633i)3-s + (0.499 − 0.866i)4-s + (−1.5 − 0.866i)5-s − 0.732i·6-s + (−2 + 3.46i)7-s − 0.999i·8-s + (1.23 + 2.13i)9-s − 1.73·10-s + 4.73·11-s + (−0.366 − 0.633i)12-s + (−5.19 − 3i)13-s + 3.99i·14-s + (−1.09 + 0.633i)15-s + (−0.5 − 0.866i)16-s + (−1.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.211 − 0.366i)3-s + (0.249 − 0.433i)4-s + (−0.670 − 0.387i)5-s − 0.298i·6-s + (−0.755 + 1.30i)7-s − 0.353i·8-s + (0.410 + 0.711i)9-s − 0.547·10-s + 1.42·11-s + (−0.105 − 0.183i)12-s + (−1.44 − 0.832i)13-s + 1.06i·14-s + (−0.283 + 0.163i)15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11787 - 0.389595i\)
\(L(\frac12)\) \(\approx\) \(1.11787 - 0.389595i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (0.5 + 6.06i)T \)
good3 \( 1 + (-0.366 + 0.633i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 + (5.19 + 3i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 0.866i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 + 0.633i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.26iT - 23T^{2} \)
29 \( 1 + 4.26iT - 29T^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
41 \( 1 + (0.232 - 0.401i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 9.46iT - 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + (-1.26 - 2.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.19 - 1.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.6 - 7.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.09 - 5.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.26 - 2.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + (7.09 + 4.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.83 + 10.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.5 + 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−14.49399322052545502238429443521, −13.07385757784209236748937081089, −12.36378321700036164341244961116, −11.65577624120736091332227575421, −9.986894660542854734188828077346, −8.811045922484158659445534165781, −7.36174003860689635232227643703, −5.88587712832323210442287172801, −4.35134283526187355372783058123, −2.51376875735871320420041919262, 3.58944060761563235987323413755, 4.35454702863015530219870428384, 6.78100534759549792262131656320, 7.12647051850906816776564857636, 9.110551508020049280683057617745, 10.19986489249041837987230741306, 11.64267762420174043955853688278, 12.52591161735497669367802514914, 13.93607461460691835680073444993, 14.62734859744811519789407056604

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