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Properties

Label 2-425-17.8-c1-0-19
Degree $2$
Conductor $425$
Sign $0.0465 + 0.998i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.70 − 1.70i)2-s + (−0.414 + i)3-s − 3.82i·4-s + (1 + 2.41i)6-s + (2.41 − i)7-s + (−3.12 − 3.12i)8-s + (1.29 + 1.29i)9-s + (−1 − 2.41i)11-s + (3.82 + 1.58i)12-s − 1.41i·13-s + (2.41 − 5.82i)14-s − 2.99·16-s + (−2.82 − 3i)17-s + 4.41·18-s + (0.585 − 0.585i)19-s + ⋯
L(s)  = 1  + (1.20 − 1.20i)2-s + (−0.239 + 0.577i)3-s − 1.91i·4-s + (0.408 + 0.985i)6-s + (0.912 − 0.377i)7-s + (−1.10 − 1.10i)8-s + (0.430 + 0.430i)9-s + (−0.301 − 0.727i)11-s + (1.10 + 0.457i)12-s − 0.392i·13-s + (0.645 − 1.55i)14-s − 0.749·16-s + (−0.685 − 0.727i)17-s + 1.04·18-s + (0.134 − 0.134i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.0465 + 0.998i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.0465 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85303 - 1.76876i\)
\(L(\frac12)\) \(\approx\) \(1.85303 - 1.76876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (2.82 + 3i)T \)
good2 \( 1 + (-1.70 + 1.70i)T - 2iT^{2} \)
3 \( 1 + (0.414 - i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-2.41 + i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1 + 2.41i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 1.41iT - 13T^{2} \)
19 \( 1 + (-0.585 + 0.585i)T - 19iT^{2} \)
23 \( 1 + (-1.82 - 4.41i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.292 + 0.121i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (3 - 7.24i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (3.53 - 8.53i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.12 + 0.464i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-0.585 - 0.585i)T + 43iT^{2} \)
47 \( 1 + 5.17iT - 47T^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + (-4.24 - 4.24i)T + 59iT^{2} \)
61 \( 1 + (3.53 - 1.46i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 1.17T + 67T^{2} \)
71 \( 1 + (2.07 - 5i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-11.9 - 4.94i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.82 - 4.41i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (8.24 - 8.24i)T - 83iT^{2} \)
89 \( 1 + 6.58iT - 89T^{2} \)
97 \( 1 + (9.53 + 3.94i)T + (68.5 + 68.5i)T^{2} \)
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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

−11.08021888094207224174930553117, −10.53749233721048542798026923408, −9.653444461161059310267285671901, −8.325838676927071518975261733381, −7.07751875889424255681743313376, −5.44161014080806992539885534764, −4.99065174586390743096056294403, −4.04065799548427285348800917714, −2.93504627725628030162268579036, −1.47240484164226113958633259240, 2.06220392461078850725746710813, 3.94999895210392187897160949399, 4.74220009891734706499106376559, 5.77460926016303749751515406702, 6.62229985583840555205685043611, 7.40091298174598342514097940813, 8.163330617793607378837060048435, 9.302710671782865051869258034272, 10.78083544650943173968502230769, 11.83714540659265136937125502087

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