Properties

Label 2-91-13.12-c1-0-4
Degree $2$
Conductor $91$
Sign $0.0862 + 0.996i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
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  − 0.688i·2-s − 2.21·3-s + 1.52·4-s − 3.21i·5-s + 1.52i·6-s i·7-s − 2.42i·8-s + 1.90·9-s − 2.21·10-s + 2.68i·11-s − 3.37·12-s + (3.59 − 0.311i)13-s − 0.688·14-s + 7.11i·15-s + 1.37·16-s − 3.59·17-s + ⋯
L(s)  = 1  − 0.487i·2-s − 1.27·3-s + 0.762·4-s − 1.43i·5-s + 0.622i·6-s − 0.377i·7-s − 0.858i·8-s + 0.634·9-s − 0.700·10-s + 0.810i·11-s − 0.975·12-s + (0.996 − 0.0862i)13-s − 0.184·14-s + 1.83i·15-s + 0.344·16-s − 0.871·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0862 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0862 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.0862 + 0.996i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.0862 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.594602 - 0.545330i\)
\(L(\frac12)\) \(\approx\) \(0.594602 - 0.545330i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + iT \)
13 \( 1 + (-3.59 + 0.311i)T \)
good2 \( 1 + 0.688iT - 2T^{2} \)
3 \( 1 + 2.21T + 3T^{2} \)
5 \( 1 + 3.21iT - 5T^{2} \)
11 \( 1 - 2.68iT - 11T^{2} \)
17 \( 1 + 3.59T + 17T^{2} \)
19 \( 1 - 8.54iT - 19T^{2} \)
23 \( 1 - 3.28T + 23T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
31 \( 1 + 5.83iT - 31T^{2} \)
37 \( 1 - 3.93iT - 37T^{2} \)
41 \( 1 + 0.755iT - 41T^{2} \)
43 \( 1 + 8.80T + 43T^{2} \)
47 \( 1 - 1.88iT - 47T^{2} \)
53 \( 1 - 2.52T + 53T^{2} \)
59 \( 1 - 7.33iT - 59T^{2} \)
61 \( 1 - 9.05T + 61T^{2} \)
67 \( 1 + 0.428iT - 67T^{2} \)
71 \( 1 + 8.98iT - 71T^{2} \)
73 \( 1 - 5.79iT - 73T^{2} \)
79 \( 1 + 4.47T + 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 - 5.36iT - 89T^{2} \)
97 \( 1 + 9.62iT - 97T^{2} \)
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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

−13.30544278620535514599108240396, −12.46352211811540891727289666759, −11.79235999144457625827927702558, −10.83035041745747949116012995888, −9.830863722000829546011011017194, −8.253813798608273811713691242534, −6.67216906210236000436164792559, −5.56018821179401057661646580728, −4.18756715711776749617986839205, −1.34673579296682142216681795402, 2.89522158791828787790384201645, 5.33654674742632997477513696031, 6.56026311914157551545955297033, 6.81074850700636120301994865983, 8.642346188122441345964472872470, 10.69031252975833843623176511870, 11.09273044815413333410432660964, 11.69373166903087370069160822320, 13.36418771109466447083758446791, 14.60120178530167868660386034700

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