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Properties

Label 2-931-7.4-c1-0-3
Degree $2$
Conductor $931$
Sign $-0.991 + 0.126i$
Analytic cond. $7.43407$
Root an. cond. $2.72654$
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  + (0.375 + 0.650i)2-s + (−1.16 + 2.01i)3-s + (0.717 − 1.24i)4-s + (−2.13 − 3.69i)5-s − 1.75·6-s + 2.58·8-s + (−1.21 − 2.10i)9-s + (1.60 − 2.77i)10-s + (−2.04 + 3.54i)11-s + (1.67 + 2.89i)12-s − 2.18·13-s + 9.94·15-s + (−0.466 − 0.808i)16-s + (−0.295 + 0.510i)17-s + (0.914 − 1.58i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.265 + 0.459i)2-s + (−0.673 + 1.16i)3-s + (0.358 − 0.621i)4-s + (−0.953 − 1.65i)5-s − 0.714·6-s + 0.912·8-s + (−0.405 − 0.703i)9-s + (0.506 − 0.877i)10-s + (−0.617 + 1.07i)11-s + (0.483 + 0.836i)12-s − 0.606·13-s + 2.56·15-s + (−0.116 − 0.202i)16-s + (−0.0715 + 0.123i)17-s + (0.215 − 0.373i)18-s + (0.114 + 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (704, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0155210 - 0.244580i\)
\(L(\frac12)\) \(\approx\) \(0.0155210 - 0.244580i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.375 - 0.650i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.16 - 2.01i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.13 + 3.69i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.04 - 3.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.18T + 13T^{2} \)
17 \( 1 + (0.295 - 0.510i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.18 - 3.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.36T + 29T^{2} \)
31 \( 1 + (2.25 - 3.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.47 + 7.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 + 8.69T + 43T^{2} \)
47 \( 1 + (-5.91 - 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.20 - 3.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.35 - 9.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.21 - 2.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.82 + 4.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.32T + 71T^{2} \)
73 \( 1 + (-5.33 + 9.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.28 + 7.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.25T + 83T^{2} \)
89 \( 1 + (1.89 + 3.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.11T + 97T^{2} \)
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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

−10.53087450484822517506033155486, −9.601995176760714432172912043798, −9.113407319584762276061700187769, −7.75278358524055467404411041959, −7.27476294747190006719053681421, −5.68934452650995156205550137770, −5.11844201039040443320275633986, −4.67648640361932658879227315379, −3.82552377078413233923923787244, −1.65399495196012351077920420198, 0.10917449168433664247225474344, 2.15913849621838037905987404340, 3.04466132089058377501540737611, 3.82733414451125363010220357371, 5.36368490532816578664045401884, 6.71301585760037029016010575808, 6.86063980417218397899750040385, 7.76229788351195735820870416487, 8.287754296493122079638870735547, 10.08082161009600188397842326360

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