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Properties

Label 2-92-92.91-c1-0-4
Degree $2$
Conductor $92$
Sign $0.625 + 0.780i$
Analytic cond. $0.734623$
Root an. cond. $0.857101$
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  + (−1 − i)2-s + i·3-s + 2i·4-s − 3.74i·5-s + (1 − i)6-s + 3.74·7-s + (2 − 2i)8-s + 2·9-s + (−3.74 + 3.74i)10-s − 3.74·11-s − 2·12-s − 13-s + (−3.74 − 3.74i)14-s + 3.74·15-s − 4·16-s + 3.74i·17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 0.577i·3-s + i·4-s − 1.67i·5-s + (0.408 − 0.408i)6-s + 1.41·7-s + (0.707 − 0.707i)8-s + 0.666·9-s + (−1.18 + 1.18i)10-s − 1.12·11-s − 0.577·12-s − 0.277·13-s + (−0.999 − 0.999i)14-s + 0.966·15-s − 16-s + 0.907i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.625 + 0.780i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 92,\ (\ :1/2),\ 0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.706597 - 0.339135i\)
\(L(\frac12)\) \(\approx\) \(0.706597 - 0.339135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
23 \( 1 + (3.74 - 3i)T \)
good3 \( 1 - iT - 3T^{2} \)
5 \( 1 + 3.74iT - 5T^{2} \)
7 \( 1 - 3.74T + 7T^{2} \)
11 \( 1 + 3.74T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 3.74iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 5iT - 31T^{2} \)
37 \( 1 - 3.74iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 7.48T + 43T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 7.48iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 3.74T + 67T^{2} \)
71 \( 1 - 5iT - 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + 7.48iT - 89T^{2} \)
97 \( 1 - 3.74iT - 97T^{2} \)
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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

−13.52933286631557637105231012063, −12.65908820192597168203023900998, −11.73774342532134154011328526395, −10.53747231230382931451715441193, −9.618615021447665228745127484935, −8.383404467759034348062976201851, −7.86073020523588095054832532109, −5.08997692610588456024831950803, −4.26217687523177261315411425500, −1.62806963452952433286508874001, 2.23664913918173950709089547972, 5.00498835510093202619703988026, 6.62028301150203796317331171312, 7.43859718267906602219076151765, 8.193847415573616937672891255914, 10.09274116297637992688470111299, 10.70587183916693670905195222688, 11.80381092297357352991568967073, 13.65174090050896348031244679519, 14.37706485636852426911476868815

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