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Properties

Label 2-74-37.10-c5-0-9
Degree $2$
Conductor $74$
Sign $-0.538 + 0.842i$
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 + (−3.37 − 5.85i)3-s + (−7.99 − 13.8i)4-s + (−12.8 − 22.3i)5-s + 27.0·6-s + (127. + 220. i)7-s + 63.9·8-s + (98.6 − 170. i)9-s + 103.·10-s − 765.·11-s + (−54.0 + 93.6i)12-s + (−283. − 490. i)13-s − 1.02e3·14-s + (−87.0 + 150. i)15-s + (−128 + 221. i)16-s + (280. − 485. i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.216 − 0.375i)3-s + (−0.249 − 0.433i)4-s + (−0.230 − 0.399i)5-s + 0.306·6-s + (0.983 + 1.70i)7-s + 0.353·8-s + (0.406 − 0.703i)9-s + 0.325·10-s − 1.90·11-s + (−0.108 + 0.187i)12-s + (−0.464 − 0.804i)13-s − 1.39·14-s + (−0.0998 + 0.173i)15-s + (−0.125 + 0.216i)16-s + (0.235 − 0.407i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.538 + 0.842i$
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: \((2,\ 74,\ (\ :5/2),\ -0.538 + 0.842i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.193888 - 0.354012i\)
\(L(\frac12)\) \(\approx\) \(0.193888 - 0.354012i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-5.46e3 + 6.28e3i)T \)
good3 \( 1 + (3.37 + 5.85i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (12.8 + 22.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-127. - 220. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 765.T + 1.61e5T^{2} \)
13 \( 1 + (283. + 490. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-280. + 485. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (616. + 1.06e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + 2.27e3T + 6.43e6T^{2} \)
29 \( 1 + 5.53e3T + 2.05e7T^{2} \)
31 \( 1 + 4.99e3T + 2.86e7T^{2} \)
41 \( 1 + (-393. - 682. i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 - 1.83e3T + 1.47e8T^{2} \)
47 \( 1 + 8.15e3T + 2.29e8T^{2} \)
53 \( 1 + (-3.15e3 + 5.45e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-579. + 1.00e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.45e4 + 2.51e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.27e4 - 5.67e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (1.80e4 + 3.12e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 3.11e4T + 2.07e9T^{2} \)
79 \( 1 + (-3.16e4 - 5.48e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-4.20e4 + 7.28e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (4.59e4 - 7.95e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 5.17e4T + 8.58e9T^{2} \)
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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

−12.97231592031468913018656039604, −12.31485357337771954101144153756, −11.03037319269180433725009220756, −9.540565050520226165893022730565, −8.352315177757493570664104444422, −7.58172585927236374882154407193, −5.78049554925256709338435582981, −5.01861280142062896565179163104, −2.30179188829408901251389137627, −0.19495538812578204086266330916, 1.85784901341971559769380227502, 3.91057075512575040287016719806, 5.00692270793826744695726789365, 7.45719923588073351404237188317, 7.913365716294367372822472716013, 9.989454001134034901990881364180, 10.65953573829505576879228262437, 11.23935154099955843193502967659, 12.91567859051911469333886294637, 13.79231721251738366346990295981

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