Properties

Degree $2$
Conductor $1925$
Sign $0.894 - 0.447i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + 2i·3-s + 4-s + 2·6-s + i·7-s − 3i·8-s − 9-s + 11-s + 2i·12-s + 4i·13-s + 14-s − 16-s − 4i·17-s + i·18-s − 2·21-s i·22-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.15i·3-s + 0.5·4-s + 0.816·6-s + 0.377i·7-s − 1.06i·8-s − 0.333·9-s + 0.301·11-s + 0.577i·12-s + 1.10i·13-s + 0.267·14-s − 0.250·16-s − 0.970i·17-s + 0.235i·18-s − 0.436·21-s − 0.213i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.894 - 0.447i$
Motivic weight: \(1\)
Character: $\chi_{1925} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.266340606\)
\(L(\frac12)\) \(\approx\) \(2.266340606\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 - iT \)
11 \( 1 - T \)
good2 \( 1 + iT - 2T^{2} \)
3 \( 1 - 2iT - 3T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.536122300748006273467764064858, −8.836199784352373422992317708260, −7.77640365622775922741520520700, −6.66106332374348458884267154469, −6.22609583639507254366492295648, −4.64462093272092201867296886648, −4.50297198453242379290658397893, −3.21988218125237988527319976309, −2.57530640345614782397191181207, −1.23714716037308604460338377373, 0.947003665188110625140996323384, 2.01156619727075697395212295152, 3.03178462562410694041578916247, 4.29041873153501702364939041195, 5.58462429807133170957624296101, 6.08293364216541482051082461888, 6.96291241772005650928043398338, 7.36584827890653499760017117079, 8.174492283481955528778622374035, 8.620955911609377922595486244277

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