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Properties

Label 1-296-296.147-r1-0-0
Degree $1$
Conductor $296$
Sign $1$
Analytic cond. $31.8096$
Root an. cond. $31.8096$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 7-s + 9-s + 11-s + 13-s + 15-s − 17-s − 19-s − 21-s + 23-s + 25-s + 27-s + 29-s + 31-s + 33-s − 35-s + 39-s + 41-s − 43-s + 45-s − 47-s + 49-s − 51-s − 53-s + 55-s − 57-s + ⋯
L(s)  = 1  + 3-s + 5-s − 7-s + 9-s + 11-s + 13-s + 15-s − 17-s − 19-s − 21-s + 23-s + 25-s + 27-s + 29-s + 31-s + 33-s − 35-s + 39-s + 41-s − 43-s + 45-s − 47-s + 49-s − 51-s − 53-s + 55-s − 57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $1$
Analytic conductor: \(31.8096\)
Root analytic conductor: \(31.8096\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{296} (147, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 296,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.485861577\)
\(L(\frac12)\) \(\approx\) \(3.485861577\)
\(L(1)\) \(\approx\) \(1.826013639\)
\(L(1)\) \(\approx\) \(1.826013639\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.12020449297753765459252930176, −24.83008700570467749672919722362, −23.37827159081759619806232273133, −22.33602458059048096879278014774, −21.49008605772180403516984968180, −20.73086118663966417638282495939, −19.67626653244875322961106069584, −19.08035281299240217398181404264, −17.99213011043261103298566396431, −16.98785286417213210159159074423, −15.940070780275119805522645290987, −15.00770638954796178906201787338, −13.96925583503559713603585898705, −13.30533633903990140931411670826, −12.587479890664771669912618948193, −10.95340659888883799532949436908, −9.89956228002693039859153846874, −9.11564107868993086927704501615, −8.45341600520253161283381473697, −6.65648871109387413682056777022, −6.39546320540671429000445011317, −4.57930731639429480174022000805, −3.42996023652599909266125860196, −2.40319843575954322389371260106, −1.18573091952039495275409350589, 1.18573091952039495275409350589, 2.40319843575954322389371260106, 3.42996023652599909266125860196, 4.57930731639429480174022000805, 6.39546320540671429000445011317, 6.65648871109387413682056777022, 8.45341600520253161283381473697, 9.11564107868993086927704501615, 9.89956228002693039859153846874, 10.95340659888883799532949436908, 12.587479890664771669912618948193, 13.30533633903990140931411670826, 13.96925583503559713603585898705, 15.00770638954796178906201787338, 15.940070780275119805522645290987, 16.98785286417213210159159074423, 17.99213011043261103298566396431, 19.08035281299240217398181404264, 19.67626653244875322961106069584, 20.73086118663966417638282495939, 21.49008605772180403516984968180, 22.33602458059048096879278014774, 23.37827159081759619806232273133, 24.83008700570467749672919722362, 25.12020449297753765459252930176

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