Properties

Label 2-1075-1.1-c1-0-42
Degree $2$
Conductor $1075$
Sign $-1$
Analytic cond. $8.58391$
Root an. cond. $2.92983$
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  − 1.96·2-s + 0.199·3-s + 1.84·4-s − 0.391·6-s + 3.80·7-s + 0.310·8-s − 2.96·9-s − 1.77·11-s + 0.368·12-s − 1.73·13-s − 7.45·14-s − 4.29·16-s + 0.964·17-s + 5.80·18-s − 4.34·19-s + 0.760·21-s + 3.48·22-s − 4.20·23-s + 0.0620·24-s + 3.40·26-s − 1.19·27-s + 7.00·28-s − 9.77·29-s + 5.61·31-s + 7.79·32-s − 0.355·33-s − 1.88·34-s + ⋯
L(s)  = 1  − 1.38·2-s + 0.115·3-s + 0.920·4-s − 0.159·6-s + 1.43·7-s + 0.109·8-s − 0.986·9-s − 0.536·11-s + 0.106·12-s − 0.481·13-s − 1.99·14-s − 1.07·16-s + 0.233·17-s + 1.36·18-s − 0.996·19-s + 0.165·21-s + 0.743·22-s − 0.876·23-s + 0.0126·24-s + 0.667·26-s − 0.229·27-s + 1.32·28-s − 1.81·29-s + 1.00·31-s + 1.37·32-s − 0.0619·33-s − 0.324·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(8.58391\)
Root analytic conductor: \(2.92983\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1075,\ (\ :1/2),\ -1)\)

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 \( 1 \)
43 \( 1 - T \)
good2 \( 1 + 1.96T + 2T^{2} \)
3 \( 1 - 0.199T + 3T^{2} \)
7 \( 1 - 3.80T + 7T^{2} \)
11 \( 1 + 1.77T + 11T^{2} \)
13 \( 1 + 1.73T + 13T^{2} \)
17 \( 1 - 0.964T + 17T^{2} \)
19 \( 1 + 4.34T + 19T^{2} \)
23 \( 1 + 4.20T + 23T^{2} \)
29 \( 1 + 9.77T + 29T^{2} \)
31 \( 1 - 5.61T + 31T^{2} \)
37 \( 1 - 0.837T + 37T^{2} \)
41 \( 1 - 7.13T + 41T^{2} \)
47 \( 1 + 7.86T + 47T^{2} \)
53 \( 1 - 1.84T + 53T^{2} \)
59 \( 1 + 8.35T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 2.64T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 4.21T + 73T^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 - 0.955T + 97T^{2} \)
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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

−9.327611568548288368033520315856, −8.586880799362715654263584911532, −7.912089733025386778165669447354, −7.58412146593238714472824966781, −6.21105432137884941802455657962, −5.18438947918664401421705406613, −4.25212141442135817195646146298, −2.55588816207873592063842635724, −1.65602883224059903134423399377, 0, 1.65602883224059903134423399377, 2.55588816207873592063842635724, 4.25212141442135817195646146298, 5.18438947918664401421705406613, 6.21105432137884941802455657962, 7.58412146593238714472824966781, 7.912089733025386778165669447354, 8.586880799362715654263584911532, 9.327611568548288368033520315856

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