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Properties

Label 2-425-85.19-c1-0-21
Degree $2$
Conductor $425$
Sign $-0.193 + 0.981i$
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  + (1.93 − 1.93i)2-s + (0.591 + 1.42i)3-s − 5.45i·4-s + (3.90 + 1.61i)6-s + (−1.16 − 0.483i)7-s + (−6.67 − 6.67i)8-s + (0.429 − 0.429i)9-s + (0.386 + 0.159i)11-s + (7.79 − 3.22i)12-s + 5.66·13-s + (−3.18 + 1.32i)14-s − 14.8·16-s + (−3.92 + 1.27i)17-s − 1.65i·18-s + (0.0948 + 0.0948i)19-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)2-s + (0.341 + 0.825i)3-s − 2.72i·4-s + (1.59 + 0.659i)6-s + (−0.441 − 0.182i)7-s + (−2.35 − 2.35i)8-s + (0.143 − 0.143i)9-s + (0.116 + 0.0482i)11-s + (2.25 − 0.932i)12-s + 1.57·13-s + (−0.852 + 0.353i)14-s − 3.71·16-s + (−0.950 + 0.309i)17-s − 0.390i·18-s + (0.0217 + 0.0217i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.193 + 0.981i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.193 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87372 - 2.28001i\)
\(L(\frac12)\) \(\approx\) \(1.87372 - 2.28001i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.92 - 1.27i)T \)
good2 \( 1 + (-1.93 + 1.93i)T - 2iT^{2} \)
3 \( 1 + (-0.591 - 1.42i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.16 + 0.483i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.386 - 0.159i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 5.66T + 13T^{2} \)
19 \( 1 + (-0.0948 - 0.0948i)T + 19iT^{2} \)
23 \( 1 + (2.60 - 6.30i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.126 - 0.304i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.559 - 0.231i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-3.38 - 8.16i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.625 + 1.50i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-1.41 - 1.41i)T + 43iT^{2} \)
47 \( 1 + 5.06T + 47T^{2} \)
53 \( 1 + (1.09 - 1.09i)T - 53iT^{2} \)
59 \( 1 + (0.997 - 0.997i)T - 59iT^{2} \)
61 \( 1 + (-2.98 + 7.21i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 + (4.45 - 1.84i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.67 - 1.10i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.00 + 1.65i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-3.10 + 3.10i)T - 83iT^{2} \)
89 \( 1 + 4.98iT - 89T^{2} \)
97 \( 1 + (0.587 - 0.243i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.10293001272975102743207319067, −10.20054105070138999762483427017, −9.614882140639042639269249938112, −8.675528767956735334943139546813, −6.64949476015889720430161109734, −5.82788953979276922888174923558, −4.56899577797588023401346565713, −3.81602922231511016603728790507, −3.14489620111451974813899800533, −1.50882499699395965356999976831, 2.46244228834928823151973530229, 3.74054763938698212188848643716, 4.72391163919588773695797396023, 6.11733759902845511673081875150, 6.48903032179216126794637747782, 7.47571632442934037525394158788, 8.301862332150716510990820386774, 9.014500330115541619439724657812, 10.86654443263437046563070009809, 11.92290721950085350747839337014

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