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Properties

Label 2-91-91.80-c1-0-1
Degree $2$
Conductor $91$
Sign $-0.333 - 0.942i$
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  + (0.554 + 2.06i)2-s + 0.0197i·3-s + (−2.24 + 1.29i)4-s + (−0.360 + 1.34i)5-s + (−0.0408 + 0.0109i)6-s + (−1.53 − 2.15i)7-s + (−0.893 − 0.893i)8-s + 2.99·9-s − 2.98·10-s + (−0.246 − 0.246i)11-s + (−0.0255 − 0.0442i)12-s + (1.32 − 3.35i)13-s + (3.61 − 4.36i)14-s + (−0.0265 − 0.00711i)15-s + (−1.23 + 2.14i)16-s + (−0.491 − 0.850i)17-s + ⋯
L(s)  = 1  + (0.392 + 1.46i)2-s + 0.0113i·3-s + (−1.12 + 0.647i)4-s + (−0.161 + 0.601i)5-s + (−0.0166 + 0.00446i)6-s + (−0.578 − 0.815i)7-s + (−0.315 − 0.315i)8-s + 0.999·9-s − 0.943·10-s + (−0.0743 − 0.0743i)11-s + (−0.00737 − 0.0127i)12-s + (0.368 − 0.929i)13-s + (0.966 − 1.16i)14-s + (−0.00685 − 0.00183i)15-s + (−0.309 + 0.535i)16-s + (−0.119 − 0.206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.333 - 0.942i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.333 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653955 + 0.925418i\)
\(L(\frac12)\) \(\approx\) \(0.653955 + 0.925418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.53 + 2.15i)T \)
13 \( 1 + (-1.32 + 3.35i)T \)
good2 \( 1 + (-0.554 - 2.06i)T + (-1.73 + i)T^{2} \)
3 \( 1 - 0.0197iT - 3T^{2} \)
5 \( 1 + (0.360 - 1.34i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.246 + 0.246i)T + 11iT^{2} \)
17 \( 1 + (0.491 + 0.850i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.25 + 3.25i)T + 19iT^{2} \)
23 \( 1 + (-2.86 - 1.65i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.941 + 1.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.81 - 0.755i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (7.91 - 2.12i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.580 - 2.16i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.47 + 3.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.5 + 2.83i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.77 + 6.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-14.7 - 3.94i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 6.45iT - 61T^{2} \)
67 \( 1 + (-7.13 + 7.13i)T - 67iT^{2} \)
71 \( 1 + (-2.43 - 9.07i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.76 - 10.3i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.890 - 1.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.33 + 8.33i)T + 83iT^{2} \)
89 \( 1 + (-3.51 - 13.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.6 + 3.39i)T + (84.0 - 48.5i)T^{2} \)
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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

−14.71319623113640128440777640405, −13.38345356382335554229218550178, −13.01594462666775903708685517352, −11.03605324683273603236664986468, −10.05049245051831913363915010443, −8.445758996789416960654728382180, −7.12553544735199808290785827491, −6.74065727029342121995678982927, −5.13018881696964633783982280071, −3.67595249827245676211376641267, 1.86512481706940484432505585212, 3.68023685756268560814648792576, 4.88652627555538926320486058996, 6.71133785597810123887076179157, 8.660765673046416998836060519471, 9.640811314129140321277883493690, 10.66952021920370107900255781821, 11.86081044455359233519776430323, 12.66708128263205389286919359254, 13.13612045295500302103656068508

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