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Properties

Label 2-1925-385.167-c0-0-2
Degree $2$
Conductor $1925$
Sign $-0.743 - 0.668i$
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  + (0.734 + 1.44i)2-s + (−0.951 + 1.30i)4-s + (0.987 − 0.156i)7-s + (−0.987 − 0.156i)8-s + (−0.951 − 0.309i)9-s + (0.309 + 0.951i)11-s + (0.951 + 1.30i)14-s + (−0.253 − 1.59i)18-s + (−1.14 + 1.14i)22-s + (−0.831 + 0.831i)23-s + (−0.734 + 1.44i)28-s + (1.53 + 1.11i)29-s + (−0.707 + 0.707i)32-s + (1.30 − 0.951i)36-s + (−0.183 − 1.16i)37-s + ⋯
L(s)  = 1  + (0.734 + 1.44i)2-s + (−0.951 + 1.30i)4-s + (0.987 − 0.156i)7-s + (−0.987 − 0.156i)8-s + (−0.951 − 0.309i)9-s + (0.309 + 0.951i)11-s + (0.951 + 1.30i)14-s + (−0.253 − 1.59i)18-s + (−1.14 + 1.14i)22-s + (−0.831 + 0.831i)23-s + (−0.734 + 1.44i)28-s + (1.53 + 1.11i)29-s + (−0.707 + 0.707i)32-s + (1.30 − 0.951i)36-s + (−0.183 − 1.16i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.743 - 0.668i$
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: \((2,\ 1925,\ (\ :0),\ -0.743 - 0.668i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.705931504\)
\(L(\frac12)\) \(\approx\) \(1.705931504\)
\(L(1)\) 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 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.734 - 1.44i)T + (-0.587 + 0.809i)T^{2} \)
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 \( 1 + (0.183 + 1.16i)T + (-0.951 + 0.309i)T^{2} \)
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 \( 1 + (0.863 + 1.69i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (1.34 + 1.34i)T + iT^{2} \)
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

−9.354029346880147904561385511043, −8.564541438223082557818282505939, −7.936978379917934992091338564861, −7.25953135634184004588799258535, −6.49582156484131964332303136636, −5.69955654647957214922370541867, −4.97095110444922551218358881525, −4.32124275900083599114818981213, −3.31753386342043690430619960582, −1.80648261157531341491282255727, 1.09538479395176803149745290969, 2.34150163870853912892270636522, 2.95774674808650582185046034117, 4.08233936623225117627601006215, 4.75813628982080481731591131869, 5.61734080107856004183361430140, 6.34048522983087502477785210462, 7.84915922339285591316589080787, 8.416174127591103339900289261126, 9.185412704376531123076181632183

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