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Properties

Label 2-75-15.2-c7-0-3
Degree $2$
Conductor $75$
Sign $-0.799 - 0.600i$
Analytic cond. $23.4288$
Root an. cond. $4.84033$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.04 + 9.04i)2-s + (−21.3 − 41.6i)3-s − 35.7i·4-s + (569. + 183. i)6-s + (−248. − 248. i)7-s + (−834. − 834. i)8-s + (−1.27e3 + 1.77e3i)9-s − 3.55e3i·11-s + (−1.48e3 + 763. i)12-s + (9.13e3 − 9.13e3i)13-s + 4.49e3·14-s + 1.96e4·16-s + (−2.46e3 + 2.46e3i)17-s + (−4.51e3 − 2.76e4i)18-s + 3.62e4i·19-s + ⋯
L(s)  = 1  + (−0.799 + 0.799i)2-s + (−0.456 − 0.889i)3-s − 0.279i·4-s + (1.07 + 0.346i)6-s + (−0.273 − 0.273i)7-s + (−0.576 − 0.576i)8-s + (−0.583 + 0.811i)9-s − 0.805i·11-s + (−0.248 + 0.127i)12-s + (1.15 − 1.15i)13-s + 0.437·14-s + 1.20·16-s + (−0.121 + 0.121i)17-s + (−0.182 − 1.11i)18-s + 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.799 - 0.600i$
Analytic conductor: \(23.4288\)
Root analytic conductor: \(4.84033\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :7/2),\ -0.799 - 0.600i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0788637 + 0.236289i\)
\(L(\frac12)\) \(\approx\) \(0.0788637 + 0.236289i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (21.3 + 41.6i)T \)
5 \( 1 \)
good2 \( 1 + (9.04 - 9.04i)T - 128iT^{2} \)
7 \( 1 + (248. + 248. i)T + 8.23e5iT^{2} \)
11 \( 1 + 3.55e3iT - 1.94e7T^{2} \)
13 \( 1 + (-9.13e3 + 9.13e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (2.46e3 - 2.46e3i)T - 4.10e8iT^{2} \)
19 \( 1 - 3.62e4iT - 8.93e8T^{2} \)
23 \( 1 + (3.95e4 + 3.95e4i)T + 3.40e9iT^{2} \)
29 \( 1 + 6.16e4T + 1.72e10T^{2} \)
31 \( 1 + 2.27e4T + 2.75e10T^{2} \)
37 \( 1 + (1.94e5 + 1.94e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 8.06e5iT - 1.94e11T^{2} \)
43 \( 1 + (6.60e5 - 6.60e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (9.11e5 - 9.11e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (-2.67e5 - 2.67e5i)T + 1.17e12iT^{2} \)
59 \( 1 - 2.00e6T + 2.48e12T^{2} \)
61 \( 1 + 2.16e6T + 3.14e12T^{2} \)
67 \( 1 + (-2.64e6 - 2.64e6i)T + 6.06e12iT^{2} \)
71 \( 1 - 1.38e6iT - 9.09e12T^{2} \)
73 \( 1 + (6.61e5 - 6.61e5i)T - 1.10e13iT^{2} \)
79 \( 1 + 2.77e6iT - 1.92e13T^{2} \)
83 \( 1 + (-4.12e6 - 4.12e6i)T + 2.71e13iT^{2} \)
89 \( 1 - 4.51e6T + 4.42e13T^{2} \)
97 \( 1 + (3.70e6 + 3.70e6i)T + 8.07e13iT^{2} \)
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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

−13.35303951174921117438975819136, −12.60903900062381434671522877648, −11.27287642992524632055380445978, −10.08399812581447654162609355367, −8.401913507000559638191155320946, −7.936140870103183492923542986546, −6.51250476532870276284744872995, −5.81146752539370476963332723152, −3.35572402762913037734499312600, −1.09876515267231553352207262968, 0.13821293094703657195952887435, 1.90154794331335288490632899206, 3.64532357974675352714408188353, 5.21510824849115115607526789749, 6.61926561704764173068565510947, 8.741556771423628186200654332422, 9.402647828611451797529288984934, 10.38405420834923464205363360073, 11.36695674331535731424496902190, 12.04766979879282657011783296779

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