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Properties

Label 2-74-37.8-c2-0-1
Degree $2$
Conductor $74$
Sign $0.834 - 0.551i$
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  + (−1.36 − 0.366i)2-s + (3.93 + 2.26i)3-s + (1.73 + i)4-s + (0.866 − 0.232i)5-s + (−4.53 − 4.53i)6-s + (−2.63 + 4.56i)7-s + (−1.99 − 2i)8-s + (5.79 + 10.0i)9-s − 1.26·10-s − 2.58i·11-s + (4.53 + 7.86i)12-s + (14.9 − 4.00i)13-s + (5.27 − 5.27i)14-s + (3.93 + 1.05i)15-s + (1.99 + 3.46i)16-s + (0.861 − 3.21i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (1.31 + 0.756i)3-s + (0.433 + 0.250i)4-s + (0.173 − 0.0464i)5-s + (−0.756 − 0.756i)6-s + (−0.376 + 0.652i)7-s + (−0.249 − 0.250i)8-s + (0.644 + 1.11i)9-s − 0.126·10-s − 0.234i·11-s + (0.378 + 0.655i)12-s + (1.14 − 0.307i)13-s + (0.376 − 0.376i)14-s + (0.262 + 0.0702i)15-s + (0.124 + 0.216i)16-s + (0.0506 − 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.834 - 0.551i$
Analytic conductor: \(2.01635\)
Root analytic conductor: \(1.41998\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1),\ 0.834 - 0.551i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.28349 + 0.385562i\)
\(L(\frac12)\) \(\approx\) \(1.28349 + 0.385562i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
37 \( 1 + (22.3 + 29.4i)T \)
good3 \( 1 + (-3.93 - 2.26i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-0.866 + 0.232i)T + (21.6 - 12.5i)T^{2} \)
7 \( 1 + (2.63 - 4.56i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + 2.58iT - 121T^{2} \)
13 \( 1 + (-14.9 + 4.00i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (-0.861 + 3.21i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (6.00 - 1.61i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (31.8 + 31.8i)T + 529iT^{2} \)
29 \( 1 + (-35.8 + 35.8i)T - 841iT^{2} \)
31 \( 1 + (17.4 - 17.4i)T - 961iT^{2} \)
41 \( 1 + (21.0 + 12.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-40.1 - 40.1i)T + 1.84e3iT^{2} \)
47 \( 1 - 14.1T + 2.20e3T^{2} \)
53 \( 1 + (26.9 + 46.7i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (21.5 - 80.4i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (13.1 + 48.9i)T + (-3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (-23.8 - 13.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (45.4 - 78.6i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 - 111. iT - 5.32e3T^{2} \)
79 \( 1 + (-108. + 29.1i)T + (5.40e3 - 3.12e3i)T^{2} \)
83 \( 1 + (-1.27 - 2.21i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-129. - 34.8i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (50.8 + 50.8i)T + 9.40e3iT^{2} \)
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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

−14.50994453556908130563752218593, −13.54035935819834047340630873732, −12.20504009243482179291777321695, −10.68081111873039826646774321206, −9.719119410537039126898425647809, −8.768452421006062942374924521394, −8.075774316198209808606469602333, −6.12016231745987118181439840793, −3.87986094719531494706329588363, −2.48773111805036411749449610908, 1.73936633839074189077699221503, 3.57959017519975297459445528918, 6.31168680562619990783718019346, 7.45907175719772883150569382256, 8.391900763929262151758255571743, 9.423797741511469308559444518026, 10.58328909218224008335632188154, 12.17243705190297192615069591427, 13.57014618980576432824995577984, 13.94718217838683792548555719193

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