Properties

Label 2-4235-1.1-c1-0-109
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  − 1.45·2-s − 2.35·3-s + 0.119·4-s + 5-s + 3.42·6-s − 7-s + 2.73·8-s + 2.54·9-s − 1.45·10-s − 0.281·12-s − 0.899·13-s + 1.45·14-s − 2.35·15-s − 4.22·16-s + 0.644·17-s − 3.71·18-s + 0.219·19-s + 0.119·20-s + 2.35·21-s + 2.75·23-s − 6.44·24-s + 25-s + 1.30·26-s + 1.06·27-s − 0.119·28-s + 0.429·29-s + 3.42·30-s + ⋯
L(s)  = 1  − 1.02·2-s − 1.36·3-s + 0.0598·4-s + 0.447·5-s + 1.40·6-s − 0.377·7-s + 0.967·8-s + 0.849·9-s − 0.460·10-s − 0.0813·12-s − 0.249·13-s + 0.389·14-s − 0.608·15-s − 1.05·16-s + 0.156·17-s − 0.874·18-s + 0.0504·19-s + 0.0267·20-s + 0.514·21-s + 0.574·23-s − 1.31·24-s + 0.200·25-s + 0.256·26-s + 0.204·27-s − 0.0226·28-s + 0.0797·29-s + 0.626·30-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: Trivial
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 \( 1 + 1.45T + 2T^{2} \)
3 \( 1 + 2.35T + 3T^{2} \)
13 \( 1 + 0.899T + 13T^{2} \)
17 \( 1 - 0.644T + 17T^{2} \)
19 \( 1 - 0.219T + 19T^{2} \)
23 \( 1 - 2.75T + 23T^{2} \)
29 \( 1 - 0.429T + 29T^{2} \)
31 \( 1 + 6.05T + 31T^{2} \)
37 \( 1 + 0.527T + 37T^{2} \)
41 \( 1 + 9.74T + 41T^{2} \)
43 \( 1 - 2.94T + 43T^{2} \)
47 \( 1 - 1.87T + 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + 1.93T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 - 6.90T + 67T^{2} \)
71 \( 1 + 0.408T + 71T^{2} \)
73 \( 1 - 8.18T + 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 - 8.13T + 83T^{2} \)
89 \( 1 + 3.73T + 89T^{2} \)
97 \( 1 + 6.55T + 97T^{2} \)
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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

−8.135647280784013470954603758594, −7.15597302012351391651811612667, −6.73817514751564128664133565609, −5.81739163308302821940225488121, −5.18936651974289352708392623363, −4.53038772581870466259053553423, −3.36869453388368774258915598453, −2.00909472192578321546940124314, −0.977007765476218454501834392000, 0, 0.977007765476218454501834392000, 2.00909472192578321546940124314, 3.36869453388368774258915598453, 4.53038772581870466259053553423, 5.18936651974289352708392623363, 5.81739163308302821940225488121, 6.73817514751564128664133565609, 7.15597302012351391651811612667, 8.135647280784013470954603758594

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