Properties

Label 2-4235-1.1-c1-0-142
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  + 0.470·2-s − 0.577·3-s − 1.77·4-s − 5-s − 0.271·6-s + 7-s − 1.77·8-s − 2.66·9-s − 0.470·10-s + 1.02·12-s − 1.19·13-s + 0.470·14-s + 0.577·15-s + 2.72·16-s + 1.08·17-s − 1.25·18-s + 6.29·19-s + 1.77·20-s − 0.577·21-s − 6.35·23-s + 1.02·24-s + 25-s − 0.563·26-s + 3.27·27-s − 1.77·28-s + 6.79·29-s + 0.271·30-s + ⋯
L(s)  = 1  + 0.332·2-s − 0.333·3-s − 0.889·4-s − 0.447·5-s − 0.110·6-s + 0.377·7-s − 0.628·8-s − 0.888·9-s − 0.148·10-s + 0.296·12-s − 0.332·13-s + 0.125·14-s + 0.149·15-s + 0.680·16-s + 0.263·17-s − 0.295·18-s + 1.44·19-s + 0.397·20-s − 0.126·21-s − 1.32·23-s + 0.209·24-s + 0.200·25-s − 0.110·26-s + 0.630·27-s − 0.336·28-s + 1.26·29-s + 0.0495·30-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 \( 1 - 0.470T + 2T^{2} \)
3 \( 1 + 0.577T + 3T^{2} \)
13 \( 1 + 1.19T + 13T^{2} \)
17 \( 1 - 1.08T + 17T^{2} \)
19 \( 1 - 6.29T + 19T^{2} \)
23 \( 1 + 6.35T + 23T^{2} \)
29 \( 1 - 6.79T + 29T^{2} \)
31 \( 1 - 9.46T + 31T^{2} \)
37 \( 1 + 8.49T + 37T^{2} \)
41 \( 1 - 9.51T + 41T^{2} \)
43 \( 1 + 5.07T + 43T^{2} \)
47 \( 1 - 3.89T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 4.91T + 59T^{2} \)
61 \( 1 + 10.6T + 61T^{2} \)
67 \( 1 + 5.09T + 67T^{2} \)
71 \( 1 + 7.24T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + 0.0718T + 79T^{2} \)
83 \( 1 + 9.16T + 83T^{2} \)
89 \( 1 + 3.20T + 89T^{2} \)
97 \( 1 + 9.05T + 97T^{2} \)
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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

−8.130287825053530702260346931256, −7.46147121916042944857286331639, −6.31084983648432492536861403985, −5.73612648642684013565159991341, −4.91346956072138449106986103680, −4.46897731621169775459569462581, −3.40281573771736381210255959023, −2.76554758369001870681166591513, −1.15056145230138208375398836377, 0, 1.15056145230138208375398836377, 2.76554758369001870681166591513, 3.40281573771736381210255959023, 4.46897731621169775459569462581, 4.91346956072138449106986103680, 5.73612648642684013565159991341, 6.31084983648432492536861403985, 7.46147121916042944857286331639, 8.130287825053530702260346931256

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