Properties

Label 2-95-95.94-c10-0-19
Degree $2$
Conductor $95$
Sign $0.632 - 0.774i$
Analytic cond. $60.3589$
Root an. cond. $7.76910$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.02e3·4-s + (1.97e3 − 2.42e3i)5-s − 8.48e3i·7-s − 5.90e4·9-s − 2.03e5·11-s + 1.04e6·16-s + 1.85e6i·17-s − 2.47e6·19-s + (−2.02e6 + 2.47e6i)20-s − 1.18e7i·23-s + (−1.96e6 − 9.56e6i)25-s + 8.69e6i·28-s + (−2.05e7 − 1.67e7i)35-s + 6.04e7·36-s + 2.03e8i·43-s + 2.08e8·44-s + ⋯
L(s)  = 1  − 4-s + (0.632 − 0.774i)5-s − 0.504i·7-s − 0.999·9-s − 1.26·11-s + 16-s + 1.30i·17-s − 19-s + (−0.632 + 0.774i)20-s − 1.83i·23-s + (−0.200 − 0.979i)25-s + 0.504i·28-s + (−0.391 − 0.319i)35-s + 0.999·36-s + 1.38i·43-s + 1.26·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.632 - 0.774i$
Analytic conductor: \(60.3589\)
Root analytic conductor: \(7.76910\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :5),\ 0.632 - 0.774i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.612044 + 0.290556i\)
\(L(\frac12)\) \(\approx\) \(0.612044 + 0.290556i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.97e3 + 2.42e3i)T \)
19 \( 1 + 2.47e6T \)
good2 \( 1 + 1.02e3T^{2} \)
3 \( 1 + 5.90e4T^{2} \)
7 \( 1 + 8.48e3iT - 2.82e8T^{2} \)
11 \( 1 + 2.03e5T + 2.59e10T^{2} \)
13 \( 1 + 1.37e11T^{2} \)
17 \( 1 - 1.85e6iT - 2.01e12T^{2} \)
23 \( 1 + 1.18e7iT - 4.14e13T^{2} \)
29 \( 1 - 4.20e14T^{2} \)
31 \( 1 - 8.19e14T^{2} \)
37 \( 1 + 4.80e15T^{2} \)
41 \( 1 - 1.34e16T^{2} \)
43 \( 1 - 2.03e8iT - 2.16e16T^{2} \)
47 \( 1 + 4.28e7iT - 5.25e16T^{2} \)
53 \( 1 + 1.74e17T^{2} \)
59 \( 1 - 5.11e17T^{2} \)
61 \( 1 - 1.60e9T + 7.13e17T^{2} \)
67 \( 1 + 1.82e18T^{2} \)
71 \( 1 - 3.25e18T^{2} \)
73 \( 1 - 2.70e9iT - 4.29e18T^{2} \)
79 \( 1 - 9.46e18T^{2} \)
83 \( 1 - 7.57e9iT - 1.55e19T^{2} \)
89 \( 1 - 3.11e19T^{2} \)
97 \( 1 + 7.37e19T^{2} \)
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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

−12.60068889380601118170171174584, −10.76794745697115588505466805548, −9.991202521767680121983389418604, −8.595756068064213234994245972359, −8.231373408226320965486263192293, −6.17217720397381422258065391455, −5.15730103531334606385314228817, −4.13208760491157723585707550606, −2.42135042513146881104795493235, −0.75020506806192037575935143114, 0.24820434471412867367521316336, 2.27361095777284880183677926131, 3.31919316396098611609328388716, 5.14378654070056368628474526800, 5.80047099553414431823568676264, 7.45176617241802238810016800197, 8.676299838494493369399258466472, 9.600166445566417064624825380321, 10.63396366213544073879169660943, 11.77719259641969960674549637429

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