Properties

Label 2-75-15.14-c8-0-0
Degree $2$
Conductor $75$
Sign $-0.799 + 0.601i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.03·2-s + (−79.6 + 14.6i)3-s − 246.·4-s + (241. − 44.3i)6-s − 59.7i·7-s + 1.52e3·8-s + (6.13e3 − 2.32e3i)9-s + 2.18e4i·11-s + (1.96e4 − 3.60e3i)12-s + 3.84e4i·13-s + 181. i·14-s + 5.85e4·16-s − 3.61e4·17-s + (−1.86e4 + 7.06e3i)18-s + 1.36e5·19-s + ⋯
L(s)  = 1  − 0.189·2-s + (−0.983 + 0.180i)3-s − 0.964·4-s + (0.186 − 0.0341i)6-s − 0.0248i·7-s + 0.372·8-s + (0.934 − 0.354i)9-s + 1.49i·11-s + (0.948 − 0.173i)12-s + 1.34i·13-s + 0.00471i·14-s + 0.893·16-s − 0.432·17-s + (−0.177 + 0.0672i)18-s + 1.04·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.601i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.799 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.799 + 0.601i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ -0.799 + 0.601i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0357454 - 0.106982i\)
\(L(\frac12)\) \(\approx\) \(0.0357454 - 0.106982i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (79.6 - 14.6i)T \)
5 \( 1 \)
good2 \( 1 + 3.03T + 256T^{2} \)
7 \( 1 + 59.7iT - 5.76e6T^{2} \)
11 \( 1 - 2.18e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.84e4iT - 8.15e8T^{2} \)
17 \( 1 + 3.61e4T + 6.97e9T^{2} \)
19 \( 1 - 1.36e5T + 1.69e10T^{2} \)
23 \( 1 + 2.35e5T + 7.83e10T^{2} \)
29 \( 1 + 8.96e5iT - 5.00e11T^{2} \)
31 \( 1 - 6.80e4T + 8.52e11T^{2} \)
37 \( 1 + 6.65e4iT - 3.51e12T^{2} \)
41 \( 1 - 3.76e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.03e6iT - 1.16e13T^{2} \)
47 \( 1 + 5.93e6T + 2.38e13T^{2} \)
53 \( 1 + 1.31e7T + 6.22e13T^{2} \)
59 \( 1 + 1.58e7iT - 1.46e14T^{2} \)
61 \( 1 + 3.51e6T + 1.91e14T^{2} \)
67 \( 1 - 1.39e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.51e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.93e7iT - 8.06e14T^{2} \)
79 \( 1 + 3.15e7T + 1.51e15T^{2} \)
83 \( 1 - 4.88e7T + 2.25e15T^{2} \)
89 \( 1 - 1.09e8iT - 3.93e15T^{2} \)
97 \( 1 + 1.16e8iT - 7.83e15T^{2} \)
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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

−13.41423688219159996737546103183, −12.33319875985444554917315900491, −11.42764180208308206876573115045, −9.901704394805658096018865923663, −9.467501481909021994410765649527, −7.72032646951668573151216519054, −6.43362573759309250564215478039, −4.89365903828101080349923827250, −4.16805003531442442697288456650, −1.57812323861695616456451819196, 0.05733503602289319729313923547, 1.01870488916351611012484876158, 3.49131916920707632686606603551, 5.10416521456531127375099649931, 5.92458399241920909926605459363, 7.62180572406165272870849898806, 8.751677771095687657492634594637, 10.13169167137704741161343867045, 10.99317060582233421133413526028, 12.26695197293387433974497223256

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