Properties

Label 1-152-152.75-r0-0-0
Degree $1$
Conductor $152$
Sign $1$
Analytic cond. $0.705885$
Root an. cond. $0.705885$
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 + 21-s − 23-s + 25-s − 27-s + 29-s + 31-s − 33-s + 35-s + 37-s − 39-s − 41-s + 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s − 59-s + ⋯
L(s)  = 1  − 3-s − 5-s − 7-s + 9-s + 11-s + 13-s + 15-s + 17-s + 21-s − 23-s + 25-s − 27-s + 29-s + 31-s − 33-s + 35-s + 37-s − 39-s − 41-s + 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s − 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $1$
Analytic conductor: \(0.705885\)
Root analytic conductor: \(0.705885\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{152} (75, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6607783624\)
\(L(\frac12)\) \(\approx\) \(0.6607783624\)
\(L(1)\) \(\approx\) \(0.6981819238\)
\(L(1)\) \(\approx\) \(0.6981819238\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
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

−27.974782612223435936815828812713, −27.36394729141557122212844072540, −26.18920732272669110058801228255, −25.03570656031738438373345491671, −23.82656002911302603490190355199, −23.03546517339361911590564687320, −22.4831171975902518081338606882, −21.323986033541699543671381978759, −19.921211032466099821880850754791, −19.07930720260468266730379929029, −18.1480462056532279261661683663, −16.75391232872452868609638625265, −16.177410472479812603270356473786, −15.27043925225149444402190729235, −13.726439311685062036019092725554, −12.3641412344125729366071751428, −11.847943451353306640368770761690, −10.67872541432986864561167177667, −9.57362478597520850000394415294, −8.10096198733225825370167169296, −6.76191662004457664971702246735, −5.99381677708181103786151993262, −4.36625746846417773525563403107, −3.417274095235402145580073033205, −0.982619117315426971064820658161, 0.982619117315426971064820658161, 3.417274095235402145580073033205, 4.36625746846417773525563403107, 5.99381677708181103786151993262, 6.76191662004457664971702246735, 8.10096198733225825370167169296, 9.57362478597520850000394415294, 10.67872541432986864561167177667, 11.847943451353306640368770761690, 12.3641412344125729366071751428, 13.726439311685062036019092725554, 15.27043925225149444402190729235, 16.177410472479812603270356473786, 16.75391232872452868609638625265, 18.1480462056532279261661683663, 19.07930720260468266730379929029, 19.921211032466099821880850754791, 21.323986033541699543671381978759, 22.4831171975902518081338606882, 23.03546517339361911590564687320, 23.82656002911302603490190355199, 25.03570656031738438373345491671, 26.18920732272669110058801228255, 27.36394729141557122212844072540, 27.974782612223435936815828812713

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