Properties

Label 2-91-91.76-c1-0-7
Degree $2$
Conductor $91$
Sign $-0.383 + 0.923i$
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  + (0.612 − 2.28i)2-s + (1.04 − 0.601i)3-s + (−3.12 − 1.80i)4-s + (−2.25 + 2.25i)5-s + (−0.736 − 2.74i)6-s + (2.48 − 0.906i)7-s + (−2.68 + 2.68i)8-s + (−0.777 + 1.34i)9-s + (3.77 + 6.54i)10-s + (4.11 + 1.10i)11-s − 4.33·12-s + (−3.48 − 0.920i)13-s + (−0.550 − 6.24i)14-s + (−0.992 + 3.70i)15-s + (0.897 + 1.55i)16-s + (−0.721 + 1.25i)17-s + ⋯
L(s)  = 1  + (0.433 − 1.61i)2-s + (0.601 − 0.347i)3-s + (−1.56 − 0.901i)4-s + (−1.00 + 1.00i)5-s + (−0.300 − 1.12i)6-s + (0.939 − 0.342i)7-s + (−0.950 + 0.950i)8-s + (−0.259 + 0.448i)9-s + (1.19 + 2.06i)10-s + (1.23 + 0.332i)11-s − 1.25·12-s + (−0.966 − 0.255i)13-s + (−0.147 − 1.66i)14-s + (−0.256 + 0.956i)15-s + (0.224 + 0.388i)16-s + (−0.175 + 0.303i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.383 + 0.923i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.383 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.683972 - 1.02431i\)
\(L(\frac12)\) \(\approx\) \(0.683972 - 1.02431i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.48 + 0.906i)T \)
13 \( 1 + (3.48 + 0.920i)T \)
good2 \( 1 + (-0.612 + 2.28i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-1.04 + 0.601i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.25 - 2.25i)T - 5iT^{2} \)
11 \( 1 + (-4.11 - 1.10i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.721 - 1.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.649 + 2.42i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.52 - 2.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.34 - 2.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.76 + 3.76i)T - 31iT^{2} \)
37 \( 1 + (0.599 + 0.160i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.04 - 1.35i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.46 + 3.15i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.68 + 4.68i)T + 47iT^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 + (-1.96 + 0.525i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.53 + 2.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.26 + 8.46i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.31 + 2.22i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-11.1 - 11.1i)T + 73iT^{2} \)
79 \( 1 - 2.01T + 79T^{2} \)
83 \( 1 + (5.11 - 5.11i)T - 83iT^{2} \)
89 \( 1 + (-1.81 + 6.76i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.62 + 6.06i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.80871026734085300842933688675, −12.41887358703533311738518153568, −11.51454563410323558206405280763, −10.97824865612824006466127761390, −9.764569292243781047603127051544, −8.223136915414181559053523190083, −7.14019264866224886340965194407, −4.64869106541590763827241828375, −3.49016281625596192744880390371, −2.09554246036616025388985983329, 4.04070673565206710650999844612, 4.82663513674673301580800067125, 6.36270154769569423452717279066, 7.88475869204372885584020562840, 8.470298727955302784016693447519, 9.332726860986784297235154302540, 11.69642153309498509245530355581, 12.38115672666857116062014604114, 14.08718170886524103249605289496, 14.51240355867536082829327987483

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