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Properties

Label 2-92-92.91-c1-0-7
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

\(L(1)\) \(\approx\) \(0.603489 - 0.289648i\)
\(L(\frac12)\) \(\approx\) \(0.603489 - 0.289648i\)
\(L(\frac{3}{2})\) 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

−13.79139756592808700644314469031, −12.81509898799787686570190640369, −12.18365734009990219137267444108, −10.20502480746488792986201362112, −9.247705733171638758499153361632, −8.498728099871941280755118620706, −7.02967586411552443769009147024, −6.09537738027108013529802472511, −4.47854774855280332112947046928, −1.17512742785760637790603110853, 2.88379729438543074857811845954, 3.87646399942521206091126630017, 6.61271282106728524039149849772, 7.26909014234119246530747526477, 9.367053502642331049008925863655, 9.825539837070753814876225330174, 10.78942035042762801911326979284, 11.76415115977466206498956713369, 13.01184896254838892039007840476, 14.20331243209216866946525903823

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