Properties

Label 2-74-37.11-c3-0-5
Degree $2$
Conductor $74$
Sign $0.563 + 0.826i$
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  + (−1.73 + i)2-s + (−2.94 + 5.10i)3-s + (1.99 − 3.46i)4-s + (−12.8 − 7.40i)5-s − 11.7i·6-s + (7.99 − 13.8i)7-s + 7.99i·8-s + (−3.87 − 6.70i)9-s + 29.6·10-s + 61.9·11-s + (11.7 + 20.4i)12-s + (−62.4 − 36.0i)13-s + 31.9i·14-s + (75.5 − 43.6i)15-s + (−8 − 13.8i)16-s + (98.0 − 56.6i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.567 + 0.982i)3-s + (0.249 − 0.433i)4-s + (−1.14 − 0.662i)5-s − 0.802i·6-s + (0.431 − 0.747i)7-s + 0.353i·8-s + (−0.143 − 0.248i)9-s + 0.936·10-s + 1.69·11-s + (0.283 + 0.491i)12-s + (−1.33 − 0.769i)13-s + 0.610i·14-s + (1.30 − 0.751i)15-s + (−0.125 − 0.216i)16-s + (1.39 − 0.807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.563 + 0.826i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ 0.563 + 0.826i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.499320 - 0.263972i\)
\(L(\frac12)\) \(\approx\) \(0.499320 - 0.263972i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 - i)T \)
37 \( 1 + (-49.0 + 219. i)T \)
good3 \( 1 + (2.94 - 5.10i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (12.8 + 7.40i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-7.99 + 13.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 - 61.9T + 1.33e3T^{2} \)
13 \( 1 + (62.4 + 36.0i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-98.0 + 56.6i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (133. + 77.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 98.0iT - 1.21e4T^{2} \)
29 \( 1 - 57.1iT - 2.43e4T^{2} \)
31 \( 1 - 21.7iT - 2.97e4T^{2} \)
41 \( 1 + (133. - 231. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + 284. iT - 7.95e4T^{2} \)
47 \( 1 + 158.T + 1.03e5T^{2} \)
53 \( 1 + (153. + 266. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-156. + 90.4i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-343. - 198. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-35.1 + 60.8i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-92.9 + 160. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 288.T + 3.89e5T^{2} \)
79 \( 1 + (-52.9 - 30.5i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-74.1 - 128. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (662. - 382. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.43e3iT - 9.12e5T^{2} \)
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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

−14.45222887843741105592584784269, −12.43056965827444773229977810198, −11.49313405868748287069392358519, −10.48983046338060948521605687380, −9.415332792985835326325813004851, −8.141257414118376736178040981585, −6.99925228251323481689453574243, −5.02287971118973092561869240459, −4.12031639365464785143344275168, −0.49808753929601882436852409760, 1.68276021394791268978948652243, 3.87151646903913092524651489283, 6.26346792265287617639888541591, 7.27246860389824269823037873902, 8.291597599881819983132256277704, 9.773585561321675518875165661426, 11.37982495894047135590050976664, 11.96539768519297398890832899576, 12.43485710507010511744463978375, 14.51792223274826823694782868211

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