Properties

Label 2-34-1.1-c1-0-0
Degree $2$
Conductor $34$
Sign $1$
Analytic cond. $0.271491$
Root an. cond. $0.521048$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(34\)    =    \(2 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.271491\)
Root analytic conductor: \(0.521048\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 34,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7492772210\)
\(L(\frac12)\) \(\approx\) \(0.7492772210\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
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 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
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

−16.63179484427491344136926072682, −15.73128059955905922020314753971, −14.21978494242446158082804784774, −12.85956463289156748627394436952, −11.95834417367800404279447796077, −10.84894668716717132123408330664, −9.286071988264369821451329617999, −6.65330731261113455462227374582, −5.99911855171913780844071238816, −3.90229547122613769308947697962, 3.90229547122613769308947697962, 5.99911855171913780844071238816, 6.65330731261113455462227374582, 9.286071988264369821451329617999, 10.84894668716717132123408330664, 11.95834417367800404279447796077, 12.85956463289156748627394436952, 14.21978494242446158082804784774, 15.73128059955905922020314753971, 16.63179484427491344136926072682

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