Properties

Label 2-175-7.6-c2-0-16
Degree $2$
Conductor $175$
Sign $1$
Analytic cond. $4.76840$
Root an. cond. $2.18366$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 5·4-s + 7·7-s + 3·8-s + 9·9-s − 6·11-s + 21·14-s − 11·16-s + 27·18-s − 18·22-s − 18·23-s + 35·28-s − 54·29-s − 45·32-s + 45·36-s + 38·37-s − 58·43-s − 30·44-s − 54·46-s + 49·49-s + 6·53-s + 21·56-s − 162·58-s + 63·63-s − 91·64-s + 118·67-s + 114·71-s + ⋯
L(s)  = 1  + 3/2·2-s + 5/4·4-s + 7-s + 3/8·8-s + 9-s − 0.545·11-s + 3/2·14-s − 0.687·16-s + 3/2·18-s − 0.818·22-s − 0.782·23-s + 5/4·28-s − 1.86·29-s − 1.40·32-s + 5/4·36-s + 1.02·37-s − 1.34·43-s − 0.681·44-s − 1.17·46-s + 49-s + 6/53·53-s + 3/8·56-s − 2.79·58-s + 63-s − 1.42·64-s + 1.76·67-s + 1.60·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(4.76840\)
Root analytic conductor: \(2.18366\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{175} (76, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.340644829\)
\(L(\frac12)\) \(\approx\) \(3.340644829\)
\(L(2)\) 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 \( 1 - p T \)
good2 \( 1 - 3 T + p^{2} T^{2} \)
3 \( ( 1 - p T )( 1 + p T ) \)
11 \( 1 + 6 T + p^{2} T^{2} \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( ( 1 - p T )( 1 + p T ) \)
23 \( 1 + 18 T + p^{2} T^{2} \)
29 \( 1 + 54 T + p^{2} T^{2} \)
31 \( ( 1 - p T )( 1 + p T ) \)
37 \( 1 - 38 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 + 58 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( 1 - 6 T + p^{2} T^{2} \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( ( 1 - p T )( 1 + p T ) \)
67 \( 1 - 118 T + p^{2} T^{2} \)
71 \( 1 - 114 T + p^{2} T^{2} \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( 1 + 94 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( ( 1 - p T )( 1 + p T ) \)
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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

−12.73499309097954401241261364186, −11.72114946429536415960283641224, −10.90396618488062604506571068582, −9.634744201767806553925033946699, −8.094351705474056676734448544284, −7.03718216062352377145727309085, −5.70021710145192853096878990643, −4.75622814773392679209724927762, −3.78207417059432049215665954029, −2.05341769902145376345960214580, 2.05341769902145376345960214580, 3.78207417059432049215665954029, 4.75622814773392679209724927762, 5.70021710145192853096878990643, 7.03718216062352377145727309085, 8.094351705474056676734448544284, 9.634744201767806553925033946699, 10.90396618488062604506571068582, 11.72114946429536415960283641224, 12.73499309097954401241261364186

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