Properties

Label 2-931-1.1-c1-0-25
Degree $2$
Conductor $931$
Sign $-1$
Analytic cond. $7.43407$
Root an. cond. $2.72654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.115·2-s − 2.98·3-s − 1.98·4-s − 0.455·5-s − 0.345·6-s − 0.462·8-s + 5.89·9-s − 0.0528·10-s + 3.02·11-s + 5.92·12-s + 0.114·13-s + 1.35·15-s + 3.91·16-s + 0.380·17-s + 0.683·18-s − 19-s + 0.904·20-s + 0.350·22-s + 7.55·23-s + 1.37·24-s − 4.79·25-s + 0.0132·26-s − 8.61·27-s − 7.78·29-s + 0.157·30-s − 6.78·31-s + 1.37·32-s + ⋯
L(s)  = 1  + 0.0820·2-s − 1.72·3-s − 0.993·4-s − 0.203·5-s − 0.141·6-s − 0.163·8-s + 1.96·9-s − 0.0167·10-s + 0.911·11-s + 1.70·12-s + 0.0316·13-s + 0.350·15-s + 0.979·16-s + 0.0921·17-s + 0.161·18-s − 0.229·19-s + 0.202·20-s + 0.0747·22-s + 1.57·23-s + 0.281·24-s − 0.958·25-s + 0.00259·26-s − 1.65·27-s − 1.44·29-s + 0.0287·30-s − 1.21·31-s + 0.243·32-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 - 0.115T + 2T^{2} \)
3 \( 1 + 2.98T + 3T^{2} \)
5 \( 1 + 0.455T + 5T^{2} \)
11 \( 1 - 3.02T + 11T^{2} \)
13 \( 1 - 0.114T + 13T^{2} \)
17 \( 1 - 0.380T + 17T^{2} \)
23 \( 1 - 7.55T + 23T^{2} \)
29 \( 1 + 7.78T + 29T^{2} \)
31 \( 1 + 6.78T + 31T^{2} \)
37 \( 1 + 1.55T + 37T^{2} \)
41 \( 1 - 6.21T + 41T^{2} \)
43 \( 1 - 9.25T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + 8.04T + 53T^{2} \)
59 \( 1 - 7.35T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 + 2.62T + 67T^{2} \)
71 \( 1 - 3.18T + 71T^{2} \)
73 \( 1 - 9.01T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 3.10T + 83T^{2} \)
89 \( 1 + 4.12T + 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−9.554971790361386462718659656404, −9.165459701562662753514903364835, −7.81201915772127649873105568476, −6.91444927042994828329576747938, −5.96081733814866279498058229310, −5.32029180463887611097034603603, −4.44917347117438314223470679550, −3.65187988550218330604301043548, −1.32929827621645345330969015361, 0, 1.32929827621645345330969015361, 3.65187988550218330604301043548, 4.44917347117438314223470679550, 5.32029180463887611097034603603, 5.96081733814866279498058229310, 6.91444927042994828329576747938, 7.81201915772127649873105568476, 9.165459701562662753514903364835, 9.554971790361386462718659656404

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