Properties

Label 2-345-1.1-c1-0-11
Degree $2$
Conductor $345$
Sign $1$
Analytic cond. $2.75483$
Root an. cond. $1.65977$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.44·2-s + 3-s + 3.99·4-s − 5-s + 2.44·6-s − 7-s + 4.89·8-s + 9-s − 2.44·10-s − 2.44·11-s + 3.99·12-s − 0.449·13-s − 2.44·14-s − 15-s + 3.99·16-s − 0.550·17-s + 2.44·18-s − 0.449·19-s − 3.99·20-s − 21-s − 5.99·22-s − 23-s + 4.89·24-s + 25-s − 1.10·26-s + 27-s − 3.99·28-s + ⋯
L(s)  = 1  + 1.73·2-s + 0.577·3-s + 1.99·4-s − 0.447·5-s + 0.999·6-s − 0.377·7-s + 1.73·8-s + 0.333·9-s − 0.774·10-s − 0.738·11-s + 1.15·12-s − 0.124·13-s − 0.654·14-s − 0.258·15-s + 0.999·16-s − 0.133·17-s + 0.577·18-s − 0.103·19-s − 0.894·20-s − 0.218·21-s − 1.27·22-s − 0.208·23-s + 0.999·24-s + 0.200·25-s − 0.215·26-s + 0.192·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(2.75483\)
Root analytic conductor: \(1.65977\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 345,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.510051679\)
\(L(\frac12)\) \(\approx\) \(3.510051679\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
23 \( 1 + T \)
good2 \( 1 - 2.44T + 2T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
13 \( 1 + 0.449T + 13T^{2} \)
17 \( 1 + 0.550T + 17T^{2} \)
19 \( 1 + 0.449T + 19T^{2} \)
29 \( 1 + 4.34T + 29T^{2} \)
31 \( 1 - 9.89T + 31T^{2} \)
37 \( 1 + 5.89T + 37T^{2} \)
41 \( 1 - 0.550T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 3.55T + 47T^{2} \)
53 \( 1 - 5.44T + 53T^{2} \)
59 \( 1 + 4.34T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 9.34T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 9.24T + 83T^{2} \)
89 \( 1 + 7.10T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−11.86077193248076654356303449390, −10.90408889797468358163310340330, −9.878449378802652331564137511855, −8.485132080751759459239239063870, −7.43318769085195749027037335195, −6.51786511086534937764587802081, −5.37555937173621067054577784413, −4.35770431936673725435082309572, −3.38598750525546634169084841382, −2.38821824858040218890160403827, 2.38821824858040218890160403827, 3.38598750525546634169084841382, 4.35770431936673725435082309572, 5.37555937173621067054577784413, 6.51786511086534937764587802081, 7.43318769085195749027037335195, 8.485132080751759459239239063870, 9.878449378802652331564137511855, 10.90408889797468358163310340330, 11.86077193248076654356303449390

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