Properties

Label 2-4235-1.1-c1-0-149
Degree $2$
Conductor $4235$
Sign $-1$
Analytic cond. $33.8166$
Root an. cond. $5.81520$
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  + 1.58·2-s − 3.01·3-s + 0.515·4-s + 5-s − 4.78·6-s − 7-s − 2.35·8-s + 6.11·9-s + 1.58·10-s − 1.55·12-s − 3.17·13-s − 1.58·14-s − 3.01·15-s − 4.76·16-s + 6.29·17-s + 9.69·18-s − 1.15·19-s + 0.515·20-s + 3.01·21-s + 5.96·23-s + 7.10·24-s + 25-s − 5.02·26-s − 9.39·27-s − 0.515·28-s − 8.32·29-s − 4.78·30-s + ⋯
L(s)  = 1  + 1.12·2-s − 1.74·3-s + 0.257·4-s + 0.447·5-s − 1.95·6-s − 0.377·7-s − 0.832·8-s + 2.03·9-s + 0.501·10-s − 0.449·12-s − 0.879·13-s − 0.423·14-s − 0.779·15-s − 1.19·16-s + 1.52·17-s + 2.28·18-s − 0.264·19-s + 0.115·20-s + 0.658·21-s + 1.24·23-s + 1.45·24-s + 0.200·25-s − 0.986·26-s − 1.80·27-s − 0.0973·28-s − 1.54·29-s − 0.874·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(33.8166\)
Root analytic conductor: \(5.81520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4235,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - 1.58T + 2T^{2} \)
3 \( 1 + 3.01T + 3T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 - 6.29T + 17T^{2} \)
19 \( 1 + 1.15T + 19T^{2} \)
23 \( 1 - 5.96T + 23T^{2} \)
29 \( 1 + 8.32T + 29T^{2} \)
31 \( 1 - 8.69T + 31T^{2} \)
37 \( 1 - 3.37T + 37T^{2} \)
41 \( 1 + 6.78T + 41T^{2} \)
43 \( 1 + 5.11T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 8.18T + 53T^{2} \)
59 \( 1 - 7.10T + 59T^{2} \)
61 \( 1 + 7.27T + 61T^{2} \)
67 \( 1 + 8.71T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + 4.48T + 73T^{2} \)
79 \( 1 - 3.30T + 79T^{2} \)
83 \( 1 - 7.14T + 83T^{2} \)
89 \( 1 - 0.654T + 89T^{2} \)
97 \( 1 - 8.73T + 97T^{2} \)
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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

−7.63394294788925215471930647594, −6.92889602027506884270509532073, −6.19162465311068253431116062959, −5.72526031829713197925234137453, −5.06928837648018263087341356769, −4.64391974113072798603675492978, −3.62495949893557746657739443652, −2.70082654331296460105221121326, −1.22860236796060971625632474926, 0, 1.22860236796060971625632474926, 2.70082654331296460105221121326, 3.62495949893557746657739443652, 4.64391974113072798603675492978, 5.06928837648018263087341356769, 5.72526031829713197925234137453, 6.19162465311068253431116062959, 6.92889602027506884270509532073, 7.63394294788925215471930647594

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