Properties

Label 2-4235-1.1-c1-0-198
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  + 2.21·2-s − 1.45·3-s + 2.89·4-s + 5-s − 3.21·6-s + 7-s + 1.97·8-s − 0.891·9-s + 2.21·10-s − 4.19·12-s − 6.19·13-s + 2.21·14-s − 1.45·15-s − 1.42·16-s + 6.75·17-s − 1.97·18-s − 6.55·19-s + 2.89·20-s − 1.45·21-s − 1.75·23-s − 2.86·24-s + 25-s − 13.7·26-s + 5.65·27-s + 2.89·28-s + 3.32·29-s − 3.21·30-s + ⋯
L(s)  = 1  + 1.56·2-s − 0.838·3-s + 1.44·4-s + 0.447·5-s − 1.31·6-s + 0.377·7-s + 0.696·8-s − 0.297·9-s + 0.699·10-s − 1.21·12-s − 1.71·13-s + 0.591·14-s − 0.374·15-s − 0.355·16-s + 1.63·17-s − 0.464·18-s − 1.50·19-s + 0.646·20-s − 0.316·21-s − 0.366·23-s − 0.584·24-s + 0.200·25-s − 2.68·26-s + 1.08·27-s + 0.546·28-s + 0.617·29-s − 0.586·30-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 \( 1 - 2.21T + 2T^{2} \)
3 \( 1 + 1.45T + 3T^{2} \)
13 \( 1 + 6.19T + 13T^{2} \)
17 \( 1 - 6.75T + 17T^{2} \)
19 \( 1 + 6.55T + 19T^{2} \)
23 \( 1 + 1.75T + 23T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 - 5.76T + 31T^{2} \)
37 \( 1 + 9.70T + 37T^{2} \)
41 \( 1 + 7.98T + 41T^{2} \)
43 \( 1 + 5.27T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 5.33T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 7.27T + 67T^{2} \)
71 \( 1 - 8.82T + 71T^{2} \)
73 \( 1 + 0.811T + 73T^{2} \)
79 \( 1 - 4.53T + 79T^{2} \)
83 \( 1 - 0.639T + 83T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + 7.25T + 97T^{2} \)
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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

−7.85528342905723096270140165286, −6.78310640796795240493887891885, −6.39612565591760217759196600212, −5.57291025766598781735184579456, −4.96549606366245252533201261184, −4.69481152837569601797396854601, −3.46109838602505941878583850629, −2.70149793061920737179052969462, −1.75752198432567331360591350244, 0, 1.75752198432567331360591350244, 2.70149793061920737179052969462, 3.46109838602505941878583850629, 4.69481152837569601797396854601, 4.96549606366245252533201261184, 5.57291025766598781735184579456, 6.39612565591760217759196600212, 6.78310640796795240493887891885, 7.85528342905723096270140165286

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