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Properties

Label 2-74-37.6-c4-0-8
Degree $2$
Conductor $74$
Sign $0.664 + 0.746i$
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 + 3.31i·3-s − 8i·4-s + (−1.68 − 1.68i)5-s + (6.62 + 6.62i)6-s + 49.9·7-s + (−16 − 16i)8-s + 70.0·9-s − 6.75·10-s − 233. i·11-s + 26.4·12-s + (101. + 101. i)13-s + (99.8 − 99.8i)14-s + (5.58 − 5.58i)15-s − 64·16-s + (−61.4 − 61.4i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + 0.367i·3-s − 0.5i·4-s + (−0.0675 − 0.0675i)5-s + (0.183 + 0.183i)6-s + 1.01·7-s + (−0.250 − 0.250i)8-s + 0.864·9-s − 0.0675·10-s − 1.93i·11-s + 0.183·12-s + (0.601 + 0.601i)13-s + (0.509 − 0.509i)14-s + (0.0248 − 0.0248i)15-s − 0.250·16-s + (−0.212 − 0.212i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.664 + 0.746i$
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: \((2,\ 74,\ (\ :2),\ 0.664 + 0.746i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.17395 - 0.975410i\)
\(L(\frac12)\) \(\approx\) \(2.17395 - 0.975410i\)
\(L(3)\) 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 \( 1 + (991. + 944. i)T \)
good3 \( 1 - 3.31iT - 81T^{2} \)
5 \( 1 + (1.68 + 1.68i)T + 625iT^{2} \)
7 \( 1 - 49.9T + 2.40e3T^{2} \)
11 \( 1 + 233. iT - 1.46e4T^{2} \)
13 \( 1 + (-101. - 101. i)T + 2.85e4iT^{2} \)
17 \( 1 + (61.4 + 61.4i)T + 8.35e4iT^{2} \)
19 \( 1 + (-284. - 284. i)T + 1.30e5iT^{2} \)
23 \( 1 + (145. + 145. i)T + 2.79e5iT^{2} \)
29 \( 1 + (26.1 - 26.1i)T - 7.07e5iT^{2} \)
31 \( 1 + (1.13e3 - 1.13e3i)T - 9.23e5iT^{2} \)
41 \( 1 - 3.30e3iT - 2.82e6T^{2} \)
43 \( 1 + (-2.14e3 - 2.14e3i)T + 3.41e6iT^{2} \)
47 \( 1 - 1.82e3T + 4.87e6T^{2} \)
53 \( 1 + 4.21e3T + 7.89e6T^{2} \)
59 \( 1 + (3.79e3 + 3.79e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (2.05e3 - 2.05e3i)T - 1.38e7iT^{2} \)
67 \( 1 + 4.31e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.16e3T + 2.54e7T^{2} \)
73 \( 1 + 1.25e3iT - 2.83e7T^{2} \)
79 \( 1 + (-614. - 614. i)T + 3.89e7iT^{2} \)
83 \( 1 - 1.18e3T + 4.74e7T^{2} \)
89 \( 1 + (-807. + 807. i)T - 6.27e7iT^{2} \)
97 \( 1 + (2.34e3 + 2.34e3i)T + 8.85e7iT^{2} \)
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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

−13.89460892279764410051581667094, −12.53958483334001731551326765239, −11.30612000514164373590548755679, −10.72041250195949801611963198000, −9.227143861488013337333246819867, −8.000360085573070538493377204414, −6.16059597918657948394866135839, −4.77775391229015804960186323102, −3.49851286255482081567946123878, −1.34373344324753864800366262603, 1.82359154153499580321544085081, 4.14410873504473352233437347480, 5.33433124805121975581850238194, 7.12775120343918874913039283579, 7.67186428944501899671111243759, 9.320900239742646771696064889265, 10.78283519199622901059460638167, 12.09409867662684639557806240657, 12.93758122230998844764809964295, 13.96515124735604308860243626484

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