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Properties

Label 2-75-5.4-c3-0-3
Degree $2$
Conductor $75$
Sign $0.894 - 0.447i$
Analytic cond. $4.42514$
Root an. cond. $2.10360$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + 3i·3-s + 7·4-s + 3·6-s + 24i·7-s − 15i·8-s − 9·9-s + 52·11-s + 21i·12-s + 22i·13-s + 24·14-s + 41·16-s + 14i·17-s + 9i·18-s + 20·19-s + ⋯
L(s)  = 1  − 0.353i·2-s + 0.577i·3-s + 0.875·4-s + 0.204·6-s + 1.29i·7-s − 0.662i·8-s − 0.333·9-s + 1.42·11-s + 0.505i·12-s + 0.469i·13-s + 0.458·14-s + 0.640·16-s + 0.199i·17-s + 0.117i·18-s + 0.241·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.76694 + 0.417119i\)
\(L(\frac12)\) \(\approx\) \(1.76694 + 0.417119i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
good2 \( 1 + iT - 8T^{2} \)
7 \( 1 - 24iT - 343T^{2} \)
11 \( 1 - 52T + 1.33e3T^{2} \)
13 \( 1 - 22iT - 2.19e3T^{2} \)
17 \( 1 - 14iT - 4.91e3T^{2} \)
19 \( 1 - 20T + 6.85e3T^{2} \)
23 \( 1 + 168iT - 1.21e4T^{2} \)
29 \( 1 + 230T + 2.43e4T^{2} \)
31 \( 1 + 288T + 2.97e4T^{2} \)
37 \( 1 - 34iT - 5.06e4T^{2} \)
41 \( 1 - 122T + 6.89e4T^{2} \)
43 \( 1 + 188iT - 7.95e4T^{2} \)
47 \( 1 + 256iT - 1.03e5T^{2} \)
53 \( 1 + 338iT - 1.48e5T^{2} \)
59 \( 1 + 100T + 2.05e5T^{2} \)
61 \( 1 - 742T + 2.26e5T^{2} \)
67 \( 1 - 84iT - 3.00e5T^{2} \)
71 \( 1 + 328T + 3.57e5T^{2} \)
73 \( 1 + 38iT - 3.89e5T^{2} \)
79 \( 1 - 240T + 4.93e5T^{2} \)
83 \( 1 - 1.21e3iT - 5.71e5T^{2} \)
89 \( 1 + 330T + 7.04e5T^{2} \)
97 \( 1 + 866iT - 9.12e5T^{2} \)
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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

−14.52912154610541017447613916515, −12.64514096100348676998269600084, −11.79770308618924995833666313426, −11.01557235888327462688866614763, −9.609030000657861722589030808892, −8.708143628134404655471590300497, −6.85847068239929167433341054136, −5.68557577106593886599653061738, −3.76663705844693705091316609957, −2.10273741856098550614338966206, 1.43267540571281902260807251747, 3.63573819053805390161471081274, 5.79388499417521171129707756801, 7.07724320989917438990120505502, 7.63775239149220370791776289306, 9.386186119479586063722899621885, 10.90465240585744396405616813900, 11.64184431800038387971689249633, 12.95217428615287008758057524727, 14.08351614955760516841510244984

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