Properties

Degree $2$
Conductor $118354$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s − 4·7-s − 8-s + 9-s − 6·11-s − 2·12-s − 2·13-s + 4·14-s + 16-s − 17-s − 18-s − 4·19-s + 8·21-s + 6·22-s + 2·24-s − 5·25-s + 2·26-s + 4·27-s − 4·28-s + 4·31-s − 32-s + 12·33-s + 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 1.74·21-s + 1.27·22-s + 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s − 0.755·28-s + 0.718·31-s − 0.176·32-s + 2.08·33-s + 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 118354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(118354\)    =    \(2 \cdot 17 \cdot 59^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{118354} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 118354,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
17 \( 1 + T \)
59 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.90008442701427, −13.34362251091140, −12.97819592754304, −12.54979946956442, −12.14081327234015, −11.57324164505906, −11.03635236220203, −10.57865621995593, −10.24466929102898, −9.742850427308515, −9.443552533192343, −8.597085462543592, −8.174312466940740, −7.593275209770476, −7.078466780380501, −6.492187858361894, −6.082581321046240, −5.746955808820688, −5.039474696413683, −4.582190019784711, −3.754313887492880, −2.936499770864329, −2.623649589388568, −1.946050999953261, −0.8011452024165587, 0, 0, 0.8011452024165587, 1.946050999953261, 2.623649589388568, 2.936499770864329, 3.754313887492880, 4.582190019784711, 5.039474696413683, 5.746955808820688, 6.082581321046240, 6.492187858361894, 7.078466780380501, 7.593275209770476, 8.174312466940740, 8.597085462543592, 9.443552533192343, 9.742850427308515, 10.24466929102898, 10.57865621995593, 11.03635236220203, 11.57324164505906, 12.14081327234015, 12.54979946956442, 12.97819592754304, 13.34362251091140, 13.90008442701427

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