Properties

Degree $2$
Conductor $175$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 7-s − 8-s + 9-s − 11-s − 14-s − 16-s + 18-s − 22-s + 23-s − 29-s + 37-s + 43-s + 46-s + 49-s − 2·53-s + 56-s − 58-s − 63-s + 64-s + 67-s − 71-s − 72-s + 74-s + 77-s − 79-s + 81-s + ⋯
L(s)  = 1  + 2-s − 7-s − 8-s + 9-s − 11-s − 14-s − 16-s + 18-s − 22-s + 23-s − 29-s + 37-s + 43-s + 46-s + 49-s − 2·53-s + 56-s − 58-s − 63-s + 64-s + 67-s − 71-s − 72-s + 74-s + 77-s − 79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{175} (76, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8869601442\)
\(L(\frac12)\) \(\approx\) \(0.8869601442\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 - T + T^{2} \)
3 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 - T + T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 - T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 + T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 - T + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−12.90962454210898858484624780878, −12.55882728637175638347345030269, −11.11325725636170649079381317752, −9.931651058410106395609982969638, −9.120960691540419640056208813700, −7.57820558602448297607061133246, −6.41858029876641888947121384092, −5.28850110961092873256574941658, −4.13099145489053729897691030505, −2.89897822964866581003696244265, 2.89897822964866581003696244265, 4.13099145489053729897691030505, 5.28850110961092873256574941658, 6.41858029876641888947121384092, 7.57820558602448297607061133246, 9.120960691540419640056208813700, 9.931651058410106395609982969638, 11.11325725636170649079381317752, 12.55882728637175638347345030269, 12.90962454210898858484624780878

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