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Properties

Label 2-74-37.36-c1-0-0
Degree $2$
Conductor $74$
Sign $-0.753 - 0.657i$
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  + i·2-s − 2.79·3-s − 4-s + 3.79i·5-s − 2.79i·6-s − 2·7-s − i·8-s + 4.79·9-s − 3.79·10-s + 3.79·11-s + 2.79·12-s − 0.791i·13-s − 2i·14-s − 10.5i·15-s + 16-s + 1.58i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.61·3-s − 0.5·4-s + 1.69i·5-s − 1.13i·6-s − 0.755·7-s − 0.353i·8-s + 1.59·9-s − 1.19·10-s + 1.14·11-s + 0.805·12-s − 0.219i·13-s − 0.534i·14-s − 2.73i·15-s + 0.250·16-s + 0.383i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.753 - 0.657i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ -0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174723 + 0.465870i\)
\(L(\frac12)\) \(\approx\) \(0.174723 + 0.465870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
37 \( 1 + (-4 + 4.58i)T \)
good3 \( 1 + 2.79T + 3T^{2} \)
5 \( 1 - 3.79iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 3.79T + 11T^{2} \)
13 \( 1 + 0.791iT - 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 - 7.58iT - 19T^{2} \)
23 \( 1 + 0.791iT - 23T^{2} \)
29 \( 1 - 0.791iT - 29T^{2} \)
31 \( 1 + 5.37iT - 31T^{2} \)
41 \( 1 - 5.20T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 1.58T + 47T^{2} \)
53 \( 1 - 7.58T + 53T^{2} \)
59 \( 1 - 7.58iT - 59T^{2} \)
61 \( 1 + 8.20iT - 61T^{2} \)
67 \( 1 + 7.37T + 67T^{2} \)
71 \( 1 - 9.16T + 71T^{2} \)
73 \( 1 - 9.37T + 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + 3.16T + 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 - 4.41iT - 97T^{2} \)
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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

−15.03541603439504904094907466770, −14.23068044913168063095737893912, −12.69090853689407699925217616715, −11.61663988693689817395718213544, −10.60372255719571875243541001959, −9.736437178033479842865496534244, −7.48824916527483325333338914894, −6.33636746403213413824536341217, −6.02007695550803456404350041039, −3.85325996899596242988396231979, 0.860193621872294310796872716453, 4.34737798094927207572296928875, 5.32315482422903812575679653540, 6.70025664105353145292165270198, 8.899194145903140863029741060291, 9.700846868135914281300537808868, 11.19778113783628699812117761311, 11.99360225742833307502145749866, 12.66999285057884744547093174536, 13.58372145412640746794111169644

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