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Properties

Label 2-74-37.6-c4-0-10
Degree $2$
Conductor $74$
Sign $-0.881 + 0.471i$
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 + 10.0i·3-s − 8i·4-s + (−28.1 − 28.1i)5-s + (20.1 + 20.1i)6-s − 38.8·7-s + (−16 − 16i)8-s − 20.4·9-s − 112.·10-s − 22.9i·11-s + 80.5·12-s + (−217. − 217. i)13-s + (−77.6 + 77.6i)14-s + (283. − 283. i)15-s − 64·16-s + (14.6 + 14.6i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + 1.11i·3-s − 0.5i·4-s + (−1.12 − 1.12i)5-s + (0.559 + 0.559i)6-s − 0.792·7-s + (−0.250 − 0.250i)8-s − 0.252·9-s − 1.12·10-s − 0.190i·11-s + 0.559·12-s + (−1.28 − 1.28i)13-s + (−0.396 + 0.396i)14-s + (1.26 − 1.26i)15-s − 0.250·16-s + (0.0507 + 0.0507i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.881 + 0.471i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ -0.881 + 0.471i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.168007 - 0.670905i\)
\(L(\frac12)\) \(\approx\) \(0.168007 - 0.670905i\)
\(L(3)\) 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 \( 1 + (-1.15e3 + 742. i)T \)
good3 \( 1 - 10.0iT - 81T^{2} \)
5 \( 1 + (28.1 + 28.1i)T + 625iT^{2} \)
7 \( 1 + 38.8T + 2.40e3T^{2} \)
11 \( 1 + 22.9iT - 1.46e4T^{2} \)
13 \( 1 + (217. + 217. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-14.6 - 14.6i)T + 8.35e4iT^{2} \)
19 \( 1 + (1.78 + 1.78i)T + 1.30e5iT^{2} \)
23 \( 1 + (1.16 + 1.16i)T + 2.79e5iT^{2} \)
29 \( 1 + (1.13e3 - 1.13e3i)T - 7.07e5iT^{2} \)
31 \( 1 + (-875. + 875. i)T - 9.23e5iT^{2} \)
41 \( 1 + 2.31e3iT - 2.82e6T^{2} \)
43 \( 1 + (-1.23e3 - 1.23e3i)T + 3.41e6iT^{2} \)
47 \( 1 + 262.T + 4.87e6T^{2} \)
53 \( 1 + 478.T + 7.89e6T^{2} \)
59 \( 1 + (1.17e3 + 1.17e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-947. + 947. i)T - 1.38e7iT^{2} \)
67 \( 1 + 5.75e3iT - 2.01e7T^{2} \)
71 \( 1 - 3.09e3T + 2.54e7T^{2} \)
73 \( 1 - 2.48e3iT - 2.83e7T^{2} \)
79 \( 1 + (5.82e3 + 5.82e3i)T + 3.89e7iT^{2} \)
83 \( 1 + 1.00e4T + 4.74e7T^{2} \)
89 \( 1 + (-9.31e3 + 9.31e3i)T - 6.27e7iT^{2} \)
97 \( 1 + (-1.56e3 - 1.56e3i)T + 8.85e7iT^{2} \)
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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

−12.90741342631164880610654589729, −12.47867828432260399912364366022, −11.16909388501038352984632709553, −10.01520608235540648597397453972, −9.123910642561317633680351033890, −7.62286911632859618469453145451, −5.41172673787661152491792974172, −4.40701308682249473476971844109, −3.30502618125434695341072048334, −0.29355083651182001917566892722, 2.63105084285629562408102229535, 4.21200169400078484732334506183, 6.44770387631431689128730021490, 7.08996100160506688671399585034, 7.86992430614852505551122906577, 9.760133800581052391671343684805, 11.50706293989265924437016833611, 12.15952276519129596748553640036, 13.22381383218457351761045553549, 14.32068754172295604396407465368

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