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Properties

Label 2-1925-385.153-c0-0-2
Degree $2$
Conductor $1925$
Sign $0.973 + 0.229i$
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  + (−1.41 − 1.41i)2-s + 3.00i·4-s + (0.707 + 0.707i)7-s + (2.82 − 2.82i)8-s + i·9-s + 11-s − 2.00i·14-s − 5.00·16-s + (1.41 − 1.41i)18-s + (−1.41 − 1.41i)22-s + (−2.12 + 2.12i)28-s + (4.24 + 4.24i)32-s − 3.00·36-s + (−1.41 + 1.41i)43-s + 3.00i·44-s + ⋯
L(s)  = 1  + (−1.41 − 1.41i)2-s + 3.00i·4-s + (0.707 + 0.707i)7-s + (2.82 − 2.82i)8-s + i·9-s + 11-s − 2.00i·14-s − 5.00·16-s + (1.41 − 1.41i)18-s + (−1.41 − 1.41i)22-s + (−2.12 + 2.12i)28-s + (4.24 + 4.24i)32-s − 3.00·36-s + (−1.41 + 1.41i)43-s + 3.00i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6230229604\)
\(L(\frac12)\) \(\approx\) \(0.6230229604\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 - T \)
good2 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
3 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.466097072893608852449598433024, −8.642693658483584119368148854681, −8.196005116483076986038276266437, −7.48312947428695568394587402212, −6.52314639917762303054789340998, −4.99991619880792306691772258842, −4.14746460831026953567807345247, −3.07013639874083411030203053624, −2.12899992147928488391066609538, −1.39271370755344685542508542038, 0.838694198854337221122114038424, 1.81263090442213304471879161873, 3.87387683452179707840371683270, 4.86950945645798910253937650209, 5.75546301668657944797067646070, 6.70660110579733194606867878720, 6.95170553679333115555786964379, 7.932872918786443942064104263771, 8.560043677217384699472335728153, 9.241529737059538645863002968076

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