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Properties

Label 2-74-37.28-c1-0-0
Degree $2$
Conductor $74$
Sign $-0.0469 - 0.998i$
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  + (−0.984 + 0.173i)2-s + (−0.266 + 1.50i)3-s + (0.939 − 0.342i)4-s + (−1.97 + 2.35i)5-s − 1.53i·6-s + (0.153 + 0.128i)7-s + (−0.866 + 0.5i)8-s + (0.613 + 0.223i)9-s + (1.53 − 2.66i)10-s + (2.17 + 3.76i)11-s + (0.266 + 1.50i)12-s + (−1.59 − 4.38i)13-s + (−0.173 − 0.0999i)14-s + (−3.03 − 3.61i)15-s + (0.766 − 0.642i)16-s + (2.32 − 6.38i)17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (−0.153 + 0.871i)3-s + (0.469 − 0.171i)4-s + (−0.885 + 1.05i)5-s − 0.625i·6-s + (0.0578 + 0.0485i)7-s + (−0.306 + 0.176i)8-s + (0.204 + 0.0744i)9-s + (0.486 − 0.843i)10-s + (0.654 + 1.13i)11-s + (0.0768 + 0.435i)12-s + (−0.443 − 1.21i)13-s + (−0.0462 − 0.0267i)14-s + (−0.783 − 0.933i)15-s + (0.191 − 0.160i)16-s + (0.563 − 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0469 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0469 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.0469 - 0.998i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ -0.0469 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424065 + 0.444454i\)
\(L(\frac12)\) \(\approx\) \(0.424065 + 0.444454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
37 \( 1 + (-5.64 - 2.27i)T \)
good3 \( 1 + (0.266 - 1.50i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (1.97 - 2.35i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.153 - 0.128i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-2.17 - 3.76i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.59 + 4.38i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.32 + 6.38i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (-4.07 - 0.719i)T + (17.8 + 6.49i)T^{2} \)
23 \( 1 + (0.896 + 0.517i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.25 + 0.727i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.10iT - 31T^{2} \)
41 \( 1 + (4.46 - 1.62i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + 0.399iT - 43T^{2} \)
47 \( 1 + (-4.10 + 7.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.65 - 7.26i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (2.69 + 3.21i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (3.60 + 9.89i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-6.67 - 5.59i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.45 + 13.9i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 - 7.27T + 73T^{2} \)
79 \( 1 + (4.04 - 4.82i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (14.4 + 5.27i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-2.06 - 2.46i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-2.07 - 1.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−15.24902585796246352183882750060, −14.26673588998674914824663144252, −12.26155199350118035078299889183, −11.32732360037952722030788115010, −10.20705265829310922244156351758, −9.593687216992280868176493881333, −7.75007070277758753158486937005, −7.00964981948651480842764594671, −4.96846762356247204586932777589, −3.23913226609077370373622393797, 1.23225079956428996471950766790, 4.05004939622091891098649431846, 6.16899747484674839346908006613, 7.53693951191976436890229194528, 8.436149927162698828121424636447, 9.541237660507926566620426363113, 11.31326530344388801655890303292, 12.03975432922997926713689429401, 12.86725990661226752807246659727, 14.20937328017049361791883262640

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