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Properties

Label 2-1925-385.13-c0-0-6
Degree $2$
Conductor $1925$
Sign $0.365 + 0.930i$
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  + (1.44 − 0.734i)2-s + (0.951 − 1.30i)4-s + (−0.156 − 0.987i)7-s + (0.156 − 0.987i)8-s + (0.951 + 0.309i)9-s + (0.309 + 0.951i)11-s + (−0.951 − 1.30i)14-s + (1.59 − 0.253i)18-s + (1.14 + 1.14i)22-s + (−0.831 − 0.831i)23-s + (−1.44 − 0.734i)28-s + (−1.53 − 1.11i)29-s + (0.707 + 0.707i)32-s + (1.30 − 0.951i)36-s + (−1.16 + 0.183i)37-s + ⋯
L(s)  = 1  + (1.44 − 0.734i)2-s + (0.951 − 1.30i)4-s + (−0.156 − 0.987i)7-s + (0.156 − 0.987i)8-s + (0.951 + 0.309i)9-s + (0.309 + 0.951i)11-s + (−0.951 − 1.30i)14-s + (1.59 − 0.253i)18-s + (1.14 + 1.14i)22-s + (−0.831 − 0.831i)23-s + (−1.44 − 0.734i)28-s + (−1.53 − 1.11i)29-s + (0.707 + 0.707i)32-s + (1.30 − 0.951i)36-s + (−1.16 + 0.183i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.365 + 0.930i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.365 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.590563350\)
\(L(\frac12)\) \(\approx\) \(2.590563350\)
\(L(1)\) 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 \( 1 + (0.156 + 0.987i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-1.44 + 0.734i)T + (0.587 - 0.809i)T^{2} \)
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 \( 1 + (0.831 + 0.831i)T + iT^{2} \)
29 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.16 - 0.183i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (0.437 - 0.437i)T - iT^{2} \)
47 \( 1 + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (-1.69 + 0.863i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (1.34 - 1.34i)T - iT^{2} \)
71 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
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

−9.779789392763575200164112275615, −8.378363894228440865517147484103, −7.33297826253138068270231771582, −6.81523867526433489124806075963, −5.82342346029554914952971386115, −4.84642904661363081190091187374, −4.14699882819255954457318747088, −3.75375956928661087581150314095, −2.37721009795189391549352311621, −1.53651404447514449800220585742, 1.84749377042024285984390833616, 3.27347199128834862887608848566, 3.73797858634347813976373450766, 4.80249286399266463694135944027, 5.65026978883896595559560081550, 6.07111100728074626313784574060, 7.00128658917868109609872639372, 7.60138248282592527377553451442, 8.714642458138295360189392276974, 9.339380524725874410460700522715

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