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Properties

Label 2-74-37.10-c5-0-13
Degree $2$
Conductor $74$
Sign $-0.559 - 0.829i$
Analytic cond. $11.8684$
Root an. cond. $3.44505$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 3.46i)2-s + (−7.16 − 12.4i)3-s + (−7.99 − 13.8i)4-s + (−51.9 − 90.0i)5-s − 57.3·6-s + (36.2 + 62.7i)7-s − 63.9·8-s + (18.8 − 32.5i)9-s − 415.·10-s − 17.7·11-s + (−114. + 198. i)12-s + (386. + 669. i)13-s + 289.·14-s + (−745. + 1.29e3i)15-s + (−128 + 221. i)16-s + (−94.5 + 163. i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.459 − 0.796i)3-s + (−0.249 − 0.433i)4-s + (−0.930 − 1.61i)5-s − 0.650·6-s + (0.279 + 0.483i)7-s − 0.353·8-s + (0.0773 − 0.134i)9-s − 1.31·10-s − 0.0442·11-s + (−0.229 + 0.398i)12-s + (0.633 + 1.09i)13-s + 0.394·14-s + (−0.855 + 1.48i)15-s + (−0.125 + 0.216i)16-s + (−0.0793 + 0.137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 - 0.829i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.559 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.559 - 0.829i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ -0.559 - 0.829i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.429611 + 0.807802i\)
\(L(\frac12)\) \(\approx\) \(0.429611 + 0.807802i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 3.46i)T \)
37 \( 1 + (3.59e3 + 7.51e3i)T \)
good3 \( 1 + (7.16 + 12.4i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (51.9 + 90.0i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-36.2 - 62.7i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 17.7T + 1.61e5T^{2} \)
13 \( 1 + (-386. - 669. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (94.5 - 163. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.20e3 + 2.09e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 - 1.36e3T + 6.43e6T^{2} \)
29 \( 1 - 2.52e3T + 2.05e7T^{2} \)
31 \( 1 + 1.51e3T + 2.86e7T^{2} \)
41 \( 1 + (-3.43e3 - 5.95e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + 1.53e4T + 1.47e8T^{2} \)
47 \( 1 + 5.79e3T + 2.29e8T^{2} \)
53 \( 1 + (-1.88e4 + 3.25e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (408. - 706. i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.88e4 + 3.26e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.93e4 - 3.34e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (1.94e4 + 3.36e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + 6.56e4T + 2.07e9T^{2} \)
79 \( 1 + (-4.11e4 - 7.12e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (4.06e4 - 7.04e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-3.37e4 + 5.83e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 2.51e4T + 8.58e9T^{2} \)
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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

−12.75527898016902807294658177288, −11.85314735365888667590553632733, −11.29446832513101631373732334705, −9.205327148304526256967448547234, −8.432266201364041153942494973296, −6.77109580803460636762704402068, −5.18475903727798887762198162105, −4.08175414913144614057735722308, −1.63607498283917230067752736752, −0.39194879306846902262971835922, 3.28283961348646012172431663210, 4.34083256798632852835175351249, 5.94642457881459869367711478588, 7.24469222864942552527800318168, 8.171354466646235933600269760776, 10.37706702474268211255045917027, 10.71550380247309800172760081212, 11.93207560118553004771479580219, 13.49883979527699228610601527325, 14.67077388807208495316223580242

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