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Properties

Label 2-155526-1.1-c1-0-48
Degree $2$
Conductor $155526$
Sign $1$
Analytic cond. $1241.88$
Root an. cond. $35.2403$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 4·5-s − 6-s + 8-s + 9-s + 4·10-s + 5·11-s − 12-s − 2·13-s − 4·15-s + 16-s + 3·17-s + 18-s + 6·19-s + 4·20-s + 5·22-s − 24-s + 11·25-s − 2·26-s − 27-s + 29-s − 4·30-s − 10·31-s + 32-s − 5·33-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 1.50·11-s − 0.288·12-s − 0.554·13-s − 1.03·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 1.37·19-s + 0.894·20-s + 1.06·22-s − 0.204·24-s + 11/5·25-s − 0.392·26-s − 0.192·27-s + 0.185·29-s − 0.730·30-s − 1.79·31-s + 0.176·32-s − 0.870·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155526\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1241.88\)
Root analytic conductor: \(35.2403\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 155526,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.978911423\)
\(L(\frac12)\) \(\approx\) \(7.978911423\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
23 \( 1 \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.32338186002969, −12.69263689996645, −12.55670050844321, −11.97049787857085, −11.34837757649311, −11.16848094463195, −10.28087669084836, −10.07504915649841, −9.538876132795964, −9.081822420524676, −8.813643673718948, −7.507383995155685, −7.418331171803428, −6.832671892774107, −6.131219337741308, −5.829639291956441, −5.575139225630541, −4.978903675438889, −4.341548344663654, −3.827646051193310, −3.050425087623756, −2.554163212901912, −1.812215607052370, −1.313018163631988, −0.8125725181401169, 0.8125725181401169, 1.313018163631988, 1.812215607052370, 2.554163212901912, 3.050425087623756, 3.827646051193310, 4.341548344663654, 4.978903675438889, 5.575139225630541, 5.829639291956441, 6.131219337741308, 6.832671892774107, 7.418331171803428, 7.507383995155685, 8.813643673718948, 9.081822420524676, 9.538876132795964, 10.07504915649841, 10.28087669084836, 11.16848094463195, 11.34837757649311, 11.97049787857085, 12.55670050844321, 12.69263689996645, 13.32338186002969

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