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Properties

Label 2-1925-385.167-c0-0-5
Degree $2$
Conductor $1925$
Sign $0.715 + 0.698i$
Analytic cond. $0.960700$
Root an. cond. $0.980153$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0949 + 0.186i)2-s + (0.562 − 0.773i)4-s + (0.987 − 0.156i)7-s + (0.403 + 0.0639i)8-s + (−0.951 − 0.309i)9-s + (0.669 − 0.743i)11-s + (0.122 + 0.169i)14-s + (−0.269 − 0.828i)16-s + (−0.0327 − 0.206i)18-s + (0.201 + 0.0541i)22-s + (−0.575 + 0.575i)23-s + (0.434 − 0.852i)28-s + (−1.20 − 0.873i)29-s + (0.417 − 0.417i)32-s + (−0.773 + 0.562i)36-s + (0.311 + 1.96i)37-s + ⋯
L(s)  = 1  + (0.0949 + 0.186i)2-s + (0.562 − 0.773i)4-s + (0.987 − 0.156i)7-s + (0.403 + 0.0639i)8-s + (−0.951 − 0.309i)9-s + (0.669 − 0.743i)11-s + (0.122 + 0.169i)14-s + (−0.269 − 0.828i)16-s + (−0.0327 − 0.206i)18-s + (0.201 + 0.0541i)22-s + (−0.575 + 0.575i)23-s + (0.434 − 0.852i)28-s + (−1.20 − 0.873i)29-s + (0.417 − 0.417i)32-s + (−0.773 + 0.562i)36-s + (0.311 + 1.96i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.715 + 0.698i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.715 + 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.473968504\)
\(L(\frac12)\) \(\approx\) \(1.473968504\)
\(L(1)\) 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 \( 1 + (-0.987 + 0.156i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
good2 \( 1 + (-0.0949 - 0.186i)T + (-0.587 + 0.809i)T^{2} \)
3 \( 1 + (0.951 + 0.309i)T^{2} \)
13 \( 1 + (-0.587 + 0.809i)T^{2} \)
17 \( 1 + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.575 - 0.575i)T - iT^{2} \)
29 \( 1 + (1.20 + 0.873i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.311 - 1.96i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.946 - 0.946i)T + iT^{2} \)
47 \( 1 + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.863 + 1.69i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.294 - 0.294i)T + iT^{2} \)
71 \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.128 - 0.395i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.587 + 0.809i)T^{2} \)
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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

−9.346462495306733988537844610995, −8.370909840086354337842445229357, −7.82211876917575469235840055372, −6.78172465425401017629381532916, −6.01760782254415769332730142772, −5.48369356337425294938521722821, −4.51597785898343185318549397033, −3.41476963844356022680143063234, −2.22764252618617216063127035461, −1.13761487324895343474985184932, 1.79572745508039148274473175319, 2.48300339461911300305584622060, 3.68238344929482535528743927628, 4.45890680835969507887628348777, 5.46276332171535032583818529414, 6.32190662546091175919421040148, 7.44072035297472655918239505376, 7.72365134643843586913158928644, 8.758977745240355020812112092680, 9.198563277323930021150786799632

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