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Properties

Label 2-74-37.10-c5-0-3
Degree $2$
Conductor $74$
Sign $0.291 - 0.956i$
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 + (12.9 + 22.4i)3-s + (−7.99 − 13.8i)4-s + (30.5 + 52.8i)5-s + 103.·6-s + (−3.03 − 5.26i)7-s − 63.9·8-s + (−213. + 369. i)9-s + 244.·10-s + 118.·11-s + (207. − 358. i)12-s + (−74.4 − 128. i)13-s − 24.2·14-s + (−789. + 1.36e3i)15-s + (−128 + 221. i)16-s + (−1.14e3 + 1.97e3i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.830 + 1.43i)3-s + (−0.249 − 0.433i)4-s + (0.545 + 0.945i)5-s + 1.17·6-s + (−0.0234 − 0.0405i)7-s − 0.353·8-s + (−0.877 + 1.52i)9-s + 0.771·10-s + 0.295·11-s + (0.415 − 0.718i)12-s + (−0.122 − 0.211i)13-s − 0.0331·14-s + (−0.906 + 1.56i)15-s + (−0.125 + 0.216i)16-s + (−0.958 + 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.291 - 0.956i$
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: \((2,\ 74,\ (\ :5/2),\ 0.291 - 0.956i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.21465 + 1.63973i\)
\(L(\frac12)\) \(\approx\) \(2.21465 + 1.63973i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3.57e3 + 7.52e3i)T \)
good3 \( 1 + (-12.9 - 22.4i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-30.5 - 52.8i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (3.03 + 5.26i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 118.T + 1.61e5T^{2} \)
13 \( 1 + (74.4 + 128. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (1.14e3 - 1.97e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (441. + 764. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 - 3.99e3T + 6.43e6T^{2} \)
29 \( 1 - 41.3T + 2.05e7T^{2} \)
31 \( 1 - 6.07e3T + 2.86e7T^{2} \)
41 \( 1 + (-2.81e3 - 4.87e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + 556.T + 1.47e8T^{2} \)
47 \( 1 + 1.78e4T + 2.29e8T^{2} \)
53 \( 1 + (-7.40e3 + 1.28e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (2.76e3 - 4.78e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.40e4 + 2.43e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.37e4 + 2.38e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (1.97e4 + 3.41e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 8.40e4T + 2.07e9T^{2} \)
79 \( 1 + (-3.06e4 - 5.31e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-5.18e3 + 8.98e3i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-5.02e3 + 8.70e3i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.30e5T + 8.58e9T^{2} \)
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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

−13.99445785564695757152287457632, −12.96102560481910301215271701611, −11.07313855175384873715373030901, −10.52297106896224754337614791075, −9.558628478357581641162378868001, −8.530629602682116413446052149854, −6.43732069594566220815428229414, −4.74098912393962648536460456879, −3.52499553987332074454955674686, −2.39194851724707821244188002263, 1.05974194036808683631942797138, 2.68388644443369732764837200874, 4.85201197535115681831597366428, 6.42514734590255660649165437372, 7.37476516819568499977193223277, 8.628404410487519017405430834295, 9.307766066110241870436573199307, 11.70913924704730773793833150102, 12.74844404351427135310990844212, 13.43555881787242226323654076885

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