Properties

Label 2-74-37.29-c2-0-4
Degree $2$
Conductor $74$
Sign $-0.998 + 0.0509i$
Analytic cond. $2.01635$
Root an. cond. $1.41998$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (−3.15 − 1.82i)3-s + (−1.73 − i)4-s + (−0.866 − 3.23i)5-s + (−3.64 + 3.64i)6-s + (−5.59 + 9.69i)7-s + (−2 + 1.99i)8-s + (2.12 + 3.68i)9-s − 4.73·10-s − 17.4i·11-s + (3.64 + 6.30i)12-s + (−5.11 − 19.1i)13-s + (11.1 + 11.1i)14-s + (−3.15 + 11.7i)15-s + (1.99 + 3.46i)16-s + (16.2 + 4.34i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−1.05 − 0.606i)3-s + (−0.433 − 0.250i)4-s + (−0.173 − 0.646i)5-s + (−0.606 + 0.606i)6-s + (−0.799 + 1.38i)7-s + (−0.250 + 0.249i)8-s + (0.236 + 0.408i)9-s − 0.473·10-s − 1.59i·11-s + (0.303 + 0.525i)12-s + (−0.393 − 1.46i)13-s + (0.799 + 0.799i)14-s + (−0.210 + 0.784i)15-s + (0.124 + 0.216i)16-s + (0.953 + 0.255i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0509i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.998 + 0.0509i$
Analytic conductor: \(2.01635\)
Root analytic conductor: \(1.41998\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1),\ -0.998 + 0.0509i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0154398 - 0.605541i\)
\(L(\frac12)\) \(\approx\) \(0.0154398 - 0.605541i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
37 \( 1 + (10.2 + 35.5i)T \)
good3 \( 1 + (3.15 + 1.82i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (0.866 + 3.23i)T + (-21.6 + 12.5i)T^{2} \)
7 \( 1 + (5.59 - 9.69i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + 17.4iT - 121T^{2} \)
13 \( 1 + (5.11 + 19.1i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (-16.2 - 4.34i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (-1.67 - 6.25i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-2.09 + 2.09i)T - 529iT^{2} \)
29 \( 1 + (28.8 + 28.8i)T + 841iT^{2} \)
31 \( 1 + (4.29 + 4.29i)T + 961iT^{2} \)
41 \( 1 + (-42.0 - 24.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (22.4 - 22.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 13.2T + 2.20e3T^{2} \)
53 \( 1 + (-41.2 - 71.5i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-4.02 - 1.07i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-23.2 + 6.22i)T + (3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (12.5 + 7.23i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-54.7 + 94.8i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 - 47.0iT - 5.32e3T^{2} \)
79 \( 1 + (39.2 + 146. i)T + (-5.40e3 + 3.12e3i)T^{2} \)
83 \( 1 + (-16.4 - 28.4i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (12.0 - 45.0i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (50.0 - 50.0i)T - 9.40e3iT^{2} \)
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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

−13.16514372684763714747347874873, −12.51550022726119648253194024384, −11.88461757181011438268827371215, −10.77424513160067610665383311119, −9.329193120890779606897583142770, −8.127433640208280479134166803398, −5.89801864592877925215464504539, −5.56999655059796748224749363126, −3.11398359967208730135267676266, −0.56107691336599234447883931808, 3.91834189605795842790369683857, 5.04819526633100799745261451548, 6.83557462880079706884859198189, 7.19687178039513793299035367873, 9.578503024773843752824981588175, 10.30501350732118148704224743446, 11.49561294533825432151775822108, 12.69777955338275398450855028115, 14.02034714700084278178347276868, 14.91303955484384026806871160897

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