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Properties

Label 2-425-85.59-c1-0-20
Degree $2$
Conductor $425$
Sign $0.420 + 0.907i$
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  + (1.09 + 1.09i)2-s + (−1.05 − 0.436i)3-s + 0.403i·4-s + (−0.676 − 1.63i)6-s + (−1.43 − 3.45i)7-s + (1.74 − 1.74i)8-s + (−1.20 − 1.20i)9-s + (0.558 + 1.34i)11-s + (0.176 − 0.425i)12-s − 6.71·13-s + (2.21 − 5.35i)14-s + 4.64·16-s + (−0.148 − 4.12i)17-s − 2.63i·18-s + (1.32 − 1.32i)19-s + ⋯
L(s)  = 1  + (0.775 + 0.775i)2-s + (−0.608 − 0.251i)3-s + 0.201i·4-s + (−0.276 − 0.666i)6-s + (−0.541 − 1.30i)7-s + (0.618 − 0.618i)8-s + (−0.400 − 0.400i)9-s + (0.168 + 0.406i)11-s + (0.0508 − 0.122i)12-s − 1.86·13-s + (0.593 − 1.43i)14-s + 1.16·16-s + (−0.0360 − 0.999i)17-s − 0.621i·18-s + (0.304 − 0.304i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.420 + 0.907i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.420 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09771 - 0.701278i\)
\(L(\frac12)\) \(\approx\) \(1.09771 - 0.701278i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.148 + 4.12i)T \)
good2 \( 1 + (-1.09 - 1.09i)T + 2iT^{2} \)
3 \( 1 + (1.05 + 0.436i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.43 + 3.45i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.558 - 1.34i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 6.71T + 13T^{2} \)
19 \( 1 + (-1.32 + 1.32i)T - 19iT^{2} \)
23 \( 1 + (-3.90 + 1.61i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-7.01 - 2.90i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-0.495 + 1.19i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-4.17 - 1.72i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.45 + 0.601i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (4.56 - 4.56i)T - 43iT^{2} \)
47 \( 1 + 2.36T + 47T^{2} \)
53 \( 1 + (4.25 + 4.25i)T + 53iT^{2} \)
59 \( 1 + (7.29 + 7.29i)T + 59iT^{2} \)
61 \( 1 + (-2.90 + 1.20i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 6.08iT - 67T^{2} \)
71 \( 1 + (1.46 - 3.53i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-5.35 + 12.9i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.916 - 2.21i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-11.9 - 11.9i)T + 83iT^{2} \)
89 \( 1 - 1.59iT - 89T^{2} \)
97 \( 1 + (-5.20 + 12.5i)T + (-68.5 - 68.5i)T^{2} \)
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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

−11.09213357973336654974431359436, −10.04947029507982603449780492858, −9.472036597995687818428963918180, −7.68392698285143165661642073863, −6.89470416842494769991885599409, −6.54725988184005947560317503462, −5.10158011776882969377330026525, −4.60282732596244169768132787193, −3.10557392937362200796658422908, −0.69088359477152935159811162609, 2.30314828961700500757052395597, 3.11046694398793517178559649531, 4.61548412762227724485115165279, 5.34850691945502050886704422436, 6.22183543803685610119265401948, 7.71548048684796282378914201725, 8.705289378073424077396278183744, 9.830250068878196499136142289524, 10.67915090052835559302469331669, 11.63792719184480188537951511056

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