Properties

Label 2-7175-1.1-c1-0-300
Degree $2$
Conductor $7175$
Sign $-1$
Analytic cond. $57.2926$
Root an. cond. $7.56919$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.20·2-s + 0.200·3-s − 0.557·4-s + 0.241·6-s − 7-s − 3.07·8-s − 2.95·9-s + 4.57·11-s − 0.112·12-s − 0.703·13-s − 1.20·14-s − 2.57·16-s − 4.25·17-s − 3.55·18-s + 8.04·19-s − 0.200·21-s + 5.49·22-s + 5.34·23-s − 0.617·24-s − 0.844·26-s − 1.19·27-s + 0.557·28-s − 5.39·29-s + 7.61·31-s + 3.05·32-s + 0.919·33-s − 5.10·34-s + ⋯
L(s)  = 1  + 0.849·2-s + 0.116·3-s − 0.278·4-s + 0.0985·6-s − 0.377·7-s − 1.08·8-s − 0.986·9-s + 1.38·11-s − 0.0323·12-s − 0.194·13-s − 0.320·14-s − 0.643·16-s − 1.03·17-s − 0.837·18-s + 1.84·19-s − 0.0438·21-s + 1.17·22-s + 1.11·23-s − 0.126·24-s − 0.165·26-s − 0.230·27-s + 0.105·28-s − 1.00·29-s + 1.36·31-s + 0.539·32-s + 0.160·33-s − 0.876·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7175\)    =    \(5^{2} \cdot 7 \cdot 41\)
Sign: $-1$
Analytic conductor: \(57.2926\)
Root analytic conductor: \(7.56919\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7175,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + T \)
41 \( 1 + T \)
good2 \( 1 - 1.20T + 2T^{2} \)
3 \( 1 - 0.200T + 3T^{2} \)
11 \( 1 - 4.57T + 11T^{2} \)
13 \( 1 + 0.703T + 13T^{2} \)
17 \( 1 + 4.25T + 17T^{2} \)
19 \( 1 - 8.04T + 19T^{2} \)
23 \( 1 - 5.34T + 23T^{2} \)
29 \( 1 + 5.39T + 29T^{2} \)
31 \( 1 - 7.61T + 31T^{2} \)
37 \( 1 + 5.19T + 37T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 7.73T + 59T^{2} \)
61 \( 1 + 2.48T + 61T^{2} \)
67 \( 1 - 3.09T + 67T^{2} \)
71 \( 1 - 5.11T + 71T^{2} \)
73 \( 1 + 4.13T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 4.90T + 83T^{2} \)
89 \( 1 - 5.97T + 89T^{2} \)
97 \( 1 + 1.45T + 97T^{2} \)
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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

−7.40887550611737803269263661382, −6.63684201957188840573471773067, −6.17633523507797928093796315559, −5.24837915900511639413050129550, −4.87305491906945793281620210990, −3.83565761714192345552058566840, −3.32220781228903562843790656921, −2.66097505671485450455245365099, −1.29710268924653848517522682015, 0, 1.29710268924653848517522682015, 2.66097505671485450455245365099, 3.32220781228903562843790656921, 3.83565761714192345552058566840, 4.87305491906945793281620210990, 5.24837915900511639413050129550, 6.17633523507797928093796315559, 6.63684201957188840573471773067, 7.40887550611737803269263661382

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