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Properties

Label 2-35e2-5.2-c0-0-0
Degree $2$
Conductor $1225$
Sign $0.899 - 0.437i$
Analytic cond. $0.611354$
Root an. cond. $0.781891$
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  + (−1.22 − 1.22i)2-s + 1.99i·4-s + (1.22 − 1.22i)8-s + i·9-s − 11-s − 0.999·16-s + (1.22 − 1.22i)18-s + (1.22 + 1.22i)22-s + (−1.22 + 1.22i)23-s + i·29-s − 1.99·36-s + (1.22 + 1.22i)37-s + (−1.22 + 1.22i)43-s − 1.99i·44-s + 2.99·46-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)2-s + 1.99i·4-s + (1.22 − 1.22i)8-s + i·9-s − 11-s − 0.999·16-s + (1.22 − 1.22i)18-s + (1.22 + 1.22i)22-s + (−1.22 + 1.22i)23-s + i·29-s − 1.99·36-s + (1.22 + 1.22i)37-s + (−1.22 + 1.22i)43-s − 1.99i·44-s + 2.99·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(0.611354\)
Root analytic conductor: \(0.781891\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (932, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ 0.899 - 0.437i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3897577134\)
\(L(\frac12)\) \(\approx\) \(0.3897577134\)
\(L(1)\) 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 \( 1 \)
good2 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
3 \( 1 - iT^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + iT - T^{2} \)
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

−9.992927765288911876168428299009, −9.480840996255004796684298910741, −8.175639597153339557932757874344, −8.126146910770123734530555379863, −7.12911639013463136000110295451, −5.70432737776668382567047966735, −4.68077485908352453628953452659, −3.39516802043105116080282867516, −2.49933563134259233916872440427, −1.53842539097858420689547349060, 0.50897020455499073638655761821, 2.28612800923728337838285302019, 3.87326402879235251066294548054, 5.15318730674635754156384285009, 6.06864411573423634846182634193, 6.59709413734572691190127769182, 7.58369017345640506555482148653, 8.172957138962982599987499769013, 8.888263754745299180339402948652, 9.741500455314076586857271922253

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