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Properties

Label 2-91-13.4-c1-0-2
Degree $2$
Conductor $91$
Sign $0.313 - 0.949i$
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  + (1.82 + 1.05i)2-s + (−1.13 + 1.95i)3-s + (1.22 + 2.12i)4-s − 3.60i·5-s + (−4.13 + 2.38i)6-s + (−0.866 + 0.5i)7-s + 0.948i·8-s + (−1.05 − 1.83i)9-s + (3.79 − 6.57i)10-s + (0.767 + 0.443i)11-s − 5.53·12-s + (−1.17 + 3.40i)13-s − 2.10·14-s + (7.05 + 4.07i)15-s + (1.44 − 2.51i)16-s + (−2.48 − 4.29i)17-s + ⋯
L(s)  = 1  + (1.29 + 0.745i)2-s + (−0.652 + 1.13i)3-s + (0.612 + 1.06i)4-s − 1.61i·5-s + (−1.68 + 0.973i)6-s + (−0.327 + 0.188i)7-s + 0.335i·8-s + (−0.352 − 0.610i)9-s + (1.20 − 2.08i)10-s + (0.231 + 0.133i)11-s − 1.59·12-s + (−0.325 + 0.945i)13-s − 0.563·14-s + (1.82 + 1.05i)15-s + (0.362 − 0.627i)16-s + (−0.601 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.313 - 0.949i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.313 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18038 + 0.853527i\)
\(L(\frac12)\) \(\approx\) \(1.18038 + 0.853527i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (1.17 - 3.40i)T \)
good2 \( 1 + (-1.82 - 1.05i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.13 - 1.95i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.60iT - 5T^{2} \)
11 \( 1 + (-0.767 - 0.443i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.48 + 4.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.06 + 1.18i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.92 - 3.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.640 - 1.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.46iT - 31T^{2} \)
37 \( 1 + (8.34 + 4.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.4 - 6.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.82 + 3.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.98iT - 47T^{2} \)
53 \( 1 - 4.92T + 53T^{2} \)
59 \( 1 + (-6.34 + 3.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.769 - 1.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.29 - 4.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.58 - 3.22i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.14iT - 73T^{2} \)
79 \( 1 - 0.757T + 79T^{2} \)
83 \( 1 + 4.76iT - 83T^{2} \)
89 \( 1 + (-3.13 - 1.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.401 - 0.231i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.30245090119271805595220182504, −13.34785658818790305941839831742, −12.32207270096983413158476659352, −11.55513022146552498576152487929, −9.765348406127732885237439298572, −8.968856041429115878976106315126, −7.04840669033812002321486487885, −5.50068031658844402984369694698, −4.89957598556280342503211753749, −3.99794790832158864949485194602, 2.37281809325816280058643494239, 3.74834103153718575203641713323, 5.82471067815063300170082587770, 6.53703218661471713830901336496, 7.74469528359523336862338933431, 10.26392173641458628449549033867, 11.03144231686742927919000163645, 11.93724755004018094311959272545, 12.81055390391385226140366309789, 13.61120391086656579978546598347

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