Properties

Label 2-4235-1.1-c1-0-165
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  + 1.93·2-s − 2.61·3-s + 1.74·4-s − 5-s − 5.05·6-s + 7-s − 0.494·8-s + 3.81·9-s − 1.93·10-s − 4.55·12-s + 2.11·13-s + 1.93·14-s + 2.61·15-s − 4.44·16-s − 5.18·17-s + 7.38·18-s + 5.34·19-s − 1.74·20-s − 2.61·21-s + 5.31·23-s + 1.28·24-s + 25-s + 4.09·26-s − 2.12·27-s + 1.74·28-s − 2.45·29-s + 5.05·30-s + ⋯
L(s)  = 1  + 1.36·2-s − 1.50·3-s + 0.872·4-s − 0.447·5-s − 2.06·6-s + 0.377·7-s − 0.174·8-s + 1.27·9-s − 0.611·10-s − 1.31·12-s + 0.586·13-s + 0.517·14-s + 0.673·15-s − 1.11·16-s − 1.25·17-s + 1.73·18-s + 1.22·19-s − 0.390·20-s − 0.569·21-s + 1.10·23-s + 0.263·24-s + 0.200·25-s + 0.803·26-s − 0.408·27-s + 0.329·28-s − 0.455·29-s + 0.922·30-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 \( 1 - 1.93T + 2T^{2} \)
3 \( 1 + 2.61T + 3T^{2} \)
13 \( 1 - 2.11T + 13T^{2} \)
17 \( 1 + 5.18T + 17T^{2} \)
19 \( 1 - 5.34T + 19T^{2} \)
23 \( 1 - 5.31T + 23T^{2} \)
29 \( 1 + 2.45T + 29T^{2} \)
31 \( 1 + 6.26T + 31T^{2} \)
37 \( 1 - 6.82T + 37T^{2} \)
41 \( 1 - 1.91T + 41T^{2} \)
43 \( 1 + 5.23T + 43T^{2} \)
47 \( 1 + 5.48T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 + 9.29T + 61T^{2} \)
67 \( 1 + 7.45T + 67T^{2} \)
71 \( 1 - 7.37T + 71T^{2} \)
73 \( 1 + 6.61T + 73T^{2} \)
79 \( 1 - 9.28T + 79T^{2} \)
83 \( 1 + 2.17T + 83T^{2} \)
89 \( 1 + 8.52T + 89T^{2} \)
97 \( 1 + 0.130T + 97T^{2} \)
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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

−7.67088796286120976802904732601, −6.91777081615080334982184512210, −6.33642542251758182280836347271, −5.59416671734659032931693776253, −5.07238780692023546950265618168, −4.47053574540703709676938498687, −3.73607862712437498669193997138, −2.76589504566288891801587903061, −1.33819516686197425488857813623, 0, 1.33819516686197425488857813623, 2.76589504566288891801587903061, 3.73607862712437498669193997138, 4.47053574540703709676938498687, 5.07238780692023546950265618168, 5.59416671734659032931693776253, 6.33642542251758182280836347271, 6.91777081615080334982184512210, 7.67088796286120976802904732601

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