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Properties

Label 2-74-37.36-c5-0-4
Degree $2$
Conductor $74$
Sign $0.592 + 0.805i$
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  − 4i·2-s − 26.9·3-s − 16·4-s + 47.3i·5-s + 107. i·6-s − 174.·7-s + 64i·8-s + 485.·9-s + 189.·10-s − 167.·11-s + 431.·12-s + 88.9i·13-s + 697. i·14-s − 1.27e3i·15-s + 256·16-s − 1.14e3i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.73·3-s − 0.5·4-s + 0.846i·5-s + 1.22i·6-s − 1.34·7-s + 0.353i·8-s + 1.99·9-s + 0.598·10-s − 0.416·11-s + 0.865·12-s + 0.145i·13-s + 0.951i·14-s − 1.46i·15-s + 0.250·16-s − 0.962i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.592 + 0.805i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ 0.592 + 0.805i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.476085 - 0.240876i\)
\(L(\frac12)\) \(\approx\) \(0.476085 - 0.240876i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
37 \( 1 + (-6.70e3 + 4.93e3i)T \)
good3 \( 1 + 26.9T + 243T^{2} \)
5 \( 1 - 47.3iT - 3.12e3T^{2} \)
7 \( 1 + 174.T + 1.68e4T^{2} \)
11 \( 1 + 167.T + 1.61e5T^{2} \)
13 \( 1 - 88.9iT - 3.71e5T^{2} \)
17 \( 1 + 1.14e3iT - 1.41e6T^{2} \)
19 \( 1 + 168. iT - 2.47e6T^{2} \)
23 \( 1 + 4.03e3iT - 6.43e6T^{2} \)
29 \( 1 - 6.47e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.77e3iT - 2.86e7T^{2} \)
41 \( 1 - 1.32e4T + 1.15e8T^{2} \)
43 \( 1 - 1.24e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.96e4T + 2.29e8T^{2} \)
53 \( 1 + 2.82e4T + 4.18e8T^{2} \)
59 \( 1 + 7.94e3iT - 7.14e8T^{2} \)
61 \( 1 + 4.25e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.34e4T + 1.35e9T^{2} \)
71 \( 1 + 5.28e4T + 1.80e9T^{2} \)
73 \( 1 - 3.34e3T + 2.07e9T^{2} \)
79 \( 1 + 7.61e4iT - 3.07e9T^{2} \)
83 \( 1 - 7.37e4T + 3.93e9T^{2} \)
89 \( 1 - 1.30e5iT - 5.58e9T^{2} \)
97 \( 1 - 2.36e4iT - 8.58e9T^{2} \)
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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

−12.83566404745073974716449201604, −12.32800404125263722977127372754, −10.98929939647325336272665854451, −10.54696651076111536969938798942, −9.413713486113688537890564318121, −7.04030328150731436405935566511, −6.20329165778133603305220366463, −4.78314697478914733701496103870, −2.97038389580453024358400496991, −0.52167034733356770914519754726, 0.68948875359544126853068770805, 4.20006546366855963735740970216, 5.61886041710148243688008788979, 6.19788422829171355755920775172, 7.60283915714420743700626658411, 9.366495712563302316926655794348, 10.32396530892000474396440827684, 11.71858291842097780906719063233, 12.79680878997618990074632892048, 13.26481879121964097345630543187

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