Properties

Label 2-4334-1.1-c1-0-61
Degree $2$
Conductor $4334$
Sign $-1$
Analytic cond. $34.6071$
Root an. cond. $5.88278$
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-s − 1.86·3-s + 4-s − 0.189·5-s + 1.86·6-s − 4.05·7-s − 8-s + 0.471·9-s + 0.189·10-s − 11-s − 1.86·12-s + 2.20·13-s + 4.05·14-s + 0.353·15-s + 16-s − 2.49·17-s − 0.471·18-s + 3.29·19-s − 0.189·20-s + 7.54·21-s + 22-s + 2.66·23-s + 1.86·24-s − 4.96·25-s − 2.20·26-s + 4.71·27-s − 4.05·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.07·3-s + 0.5·4-s − 0.0847·5-s + 0.760·6-s − 1.53·7-s − 0.353·8-s + 0.157·9-s + 0.0599·10-s − 0.301·11-s − 0.537·12-s + 0.611·13-s + 1.08·14-s + 0.0912·15-s + 0.250·16-s − 0.604·17-s − 0.111·18-s + 0.756·19-s − 0.0423·20-s + 1.64·21-s + 0.213·22-s + 0.556·23-s + 0.380·24-s − 0.992·25-s − 0.432·26-s + 0.906·27-s − 0.765·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4334\)    =    \(2 \cdot 11 \cdot 197\)
Sign: $-1$
Analytic conductor: \(34.6071\)
Root analytic conductor: \(5.88278\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4334,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
11 \( 1 + T \)
197 \( 1 + T \)
good3 \( 1 + 1.86T + 3T^{2} \)
5 \( 1 + 0.189T + 5T^{2} \)
7 \( 1 + 4.05T + 7T^{2} \)
13 \( 1 - 2.20T + 13T^{2} \)
17 \( 1 + 2.49T + 17T^{2} \)
19 \( 1 - 3.29T + 19T^{2} \)
23 \( 1 - 2.66T + 23T^{2} \)
29 \( 1 + 6.69T + 29T^{2} \)
31 \( 1 - 0.251T + 31T^{2} \)
37 \( 1 - 7.30T + 37T^{2} \)
41 \( 1 - 3.69T + 41T^{2} \)
43 \( 1 + 2.58T + 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 - 6.06T + 53T^{2} \)
59 \( 1 - 3.76T + 59T^{2} \)
61 \( 1 + 15.4T + 61T^{2} \)
67 \( 1 - 4.97T + 67T^{2} \)
71 \( 1 - 5.09T + 71T^{2} \)
73 \( 1 - 7.97T + 73T^{2} \)
79 \( 1 + 1.72T + 79T^{2} \)
83 \( 1 - 2.47T + 83T^{2} \)
89 \( 1 - 3.84T + 89T^{2} \)
97 \( 1 + 5.55T + 97T^{2} \)
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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

−7.936445393654496996052523703415, −7.18745093282025225469005263508, −6.51630773846702734097498016214, −5.91113016094717690379330690472, −5.42451760612721858173791063454, −4.16145694394435043047600564870, −3.29179963708481230734053001480, −2.39698625252297095377258898750, −0.912859137507304723934639991040, 0, 0.912859137507304723934639991040, 2.39698625252297095377258898750, 3.29179963708481230734053001480, 4.16145694394435043047600564870, 5.42451760612721858173791063454, 5.91113016094717690379330690472, 6.51630773846702734097498016214, 7.18745093282025225469005263508, 7.936445393654496996052523703415

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