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Properties

Label 2-95-95.37-c1-0-2
Degree $2$
Conductor $95$
Sign $0.946 - 0.322i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·4-s + (0.5 − 2.17i)5-s + (3.67 + 3.67i)7-s − 3i·9-s − 4.35·11-s − 4·16-s + (−1.32 − 1.32i)17-s − 4.35i·19-s + (4.35 + i)20-s + (−2.35 + 2.35i)23-s + (−4.50 − 2.17i)25-s + (−7.35 + 7.35i)28-s + (9.85 − 6.17i)35-s + 6·36-s + (6.03 − 6.03i)43-s − 8.71i·44-s + ⋯
L(s)  = 1  + i·4-s + (0.223 − 0.974i)5-s + (1.39 + 1.39i)7-s − i·9-s − 1.31·11-s − 16-s + (−0.320 − 0.320i)17-s − 0.999i·19-s + (0.974 + 0.223i)20-s + (−0.491 + 0.491i)23-s + (−0.900 − 0.435i)25-s + (−1.39 + 1.39i)28-s + (1.66 − 1.04i)35-s + 36-s + (0.920 − 0.920i)43-s − 1.31i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.946 - 0.322i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.946 - 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04819 + 0.173494i\)
\(L(\frac12)\) \(\approx\) \(1.04819 + 0.173494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 2.17i)T \)
19 \( 1 + 4.35iT \)
good2 \( 1 - 2iT^{2} \)
3 \( 1 + 3iT^{2} \)
7 \( 1 + (-3.67 - 3.67i)T + 7iT^{2} \)
11 \( 1 + 4.35T + 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (1.32 + 1.32i)T + 17iT^{2} \)
23 \( 1 + (2.35 - 2.35i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-6.03 + 6.03i)T - 43iT^{2} \)
47 \( 1 + (-8.67 - 8.67i)T + 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 4.35T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-1.03 + 1.03i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (12.3 - 12.3i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−13.89249883825589597130202178650, −12.73494610103650727299396923874, −12.10460677653280772083684210827, −11.21879974514311920469653283152, −9.236896310080760273615181007332, −8.595687660082973392946260420401, −7.62839557848686176267533362570, −5.65352716784949934292279060435, −4.56905231088985048589981353282, −2.47726570170725241576151576310, 2.02437782406743756974955172037, 4.47109546340112711639901499806, 5.67758961533649182983175749067, 7.26104942526167636282423958242, 8.095358560424920867077006741811, 10.31401479793250400873113494884, 10.47648630082875479060343342601, 11.29973469349593288745452202926, 13.35876832377629100862708365692, 14.07381555054406990172620756617

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