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Properties

Label 2-74-37.31-c4-0-5
Degree $2$
Conductor $74$
Sign $0.817 + 0.576i$
Analytic cond. $7.64937$
Root an. cond. $2.76575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s + 0.773i·3-s + 8i·4-s + (2.70 − 2.70i)5-s + (1.54 − 1.54i)6-s − 17.1·7-s + (16 − 16i)8-s + 80.4·9-s − 10.8·10-s + 132. i·11-s − 6.19·12-s + (181. − 181. i)13-s + (34.3 + 34.3i)14-s + (2.09 + 2.09i)15-s − 64·16-s + (261. − 261. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.0859i·3-s + 0.5i·4-s + (0.108 − 0.108i)5-s + (0.0429 − 0.0429i)6-s − 0.350·7-s + (0.250 − 0.250i)8-s + 0.992·9-s − 0.108·10-s + 1.09i·11-s − 0.0429·12-s + (1.07 − 1.07i)13-s + (0.175 + 0.175i)14-s + (0.00931 + 0.00931i)15-s − 0.250·16-s + (0.903 − 0.903i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.817 + 0.576i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ 0.817 + 0.576i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.32674 - 0.421048i\)
\(L(\frac12)\) \(\approx\) \(1.32674 - 0.421048i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
37 \( 1 + (-1.04e3 - 878. i)T \)
good3 \( 1 - 0.773iT - 81T^{2} \)
5 \( 1 + (-2.70 + 2.70i)T - 625iT^{2} \)
7 \( 1 + 17.1T + 2.40e3T^{2} \)
11 \( 1 - 132. iT - 1.46e4T^{2} \)
13 \( 1 + (-181. + 181. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-261. + 261. i)T - 8.35e4iT^{2} \)
19 \( 1 + (-137. + 137. i)T - 1.30e5iT^{2} \)
23 \( 1 + (-543. + 543. i)T - 2.79e5iT^{2} \)
29 \( 1 + (624. + 624. i)T + 7.07e5iT^{2} \)
31 \( 1 + (-833. - 833. i)T + 9.23e5iT^{2} \)
41 \( 1 - 955. iT - 2.82e6T^{2} \)
43 \( 1 + (1.50e3 - 1.50e3i)T - 3.41e6iT^{2} \)
47 \( 1 + 3.09e3T + 4.87e6T^{2} \)
53 \( 1 - 4.40e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.82e3 + 1.82e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (4.09e3 + 4.09e3i)T + 1.38e7iT^{2} \)
67 \( 1 - 2.45e3iT - 2.01e7T^{2} \)
71 \( 1 - 6.68e3T + 2.54e7T^{2} \)
73 \( 1 + 3.24e3iT - 2.83e7T^{2} \)
79 \( 1 + (1.77e3 - 1.77e3i)T - 3.89e7iT^{2} \)
83 \( 1 + 5.35e3T + 4.74e7T^{2} \)
89 \( 1 + (6.27e3 + 6.27e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (-430. + 430. i)T - 8.85e7iT^{2} \)
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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

−13.27088532566450145569930593926, −12.72597678077255356661674085320, −11.41036617475761717597130712643, −10.15161654036470418325526615624, −9.496724443074630454880056303663, −7.994202905548809403846865865794, −6.80787885840979539210374435275, −4.87530576357421164647700823795, −3.16525530725461329986226375207, −1.14663205384707948046140648180, 1.29377642749412454911514237772, 3.77284861472737160353995953786, 5.76714803682980606110586781221, 6.82769633863013675809162223404, 8.153665946191621605054828657364, 9.315340366937756000031492517840, 10.41129962505428475474988262808, 11.56829721531951544236123193744, 13.08053271205744236376392377750, 13.94410050770965693258840143470

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