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Properties

Label 2-425-17.13-c1-0-9
Degree $2$
Conductor $425$
Sign $-0.323 - 0.946i$
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  + 1.44i·2-s + (1.14 + 1.14i)3-s − 0.0943·4-s + (−1.65 + 1.65i)6-s + (2.69 − 2.69i)7-s + 2.75i·8-s − 0.370i·9-s + (−4.34 + 4.34i)11-s + (−0.108 − 0.108i)12-s + 2.56·13-s + (3.90 + 3.90i)14-s − 4.17·16-s + (3.45 + 2.25i)17-s + 0.535·18-s − 1.17i·19-s + ⋯
L(s)  = 1  + 1.02i·2-s + (0.662 + 0.662i)3-s − 0.0471·4-s + (−0.677 + 0.677i)6-s + (1.01 − 1.01i)7-s + 0.975i·8-s − 0.123i·9-s + (−1.31 + 1.31i)11-s + (−0.0312 − 0.0312i)12-s + 0.712·13-s + (1.04 + 1.04i)14-s − 1.04·16-s + (0.837 + 0.546i)17-s + 0.126·18-s − 0.268i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15276 + 1.61207i\)
\(L(\frac12)\) \(\approx\) \(1.15276 + 1.61207i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.45 - 2.25i)T \)
good2 \( 1 - 1.44iT - 2T^{2} \)
3 \( 1 + (-1.14 - 1.14i)T + 3iT^{2} \)
7 \( 1 + (-2.69 + 2.69i)T - 7iT^{2} \)
11 \( 1 + (4.34 - 4.34i)T - 11iT^{2} \)
13 \( 1 - 2.56T + 13T^{2} \)
19 \( 1 + 1.17iT - 19T^{2} \)
23 \( 1 + (2.53 - 2.53i)T - 23iT^{2} \)
29 \( 1 + (3.70 + 3.70i)T + 29iT^{2} \)
31 \( 1 + (0.394 + 0.394i)T + 31iT^{2} \)
37 \( 1 + (5.28 + 5.28i)T + 37iT^{2} \)
41 \( 1 + (-5.35 + 5.35i)T - 41iT^{2} \)
43 \( 1 + 0.774iT - 43T^{2} \)
47 \( 1 + 4.35T + 47T^{2} \)
53 \( 1 + 2.36iT - 53T^{2} \)
59 \( 1 + 12.8iT - 59T^{2} \)
61 \( 1 + (3.29 - 3.29i)T - 61iT^{2} \)
67 \( 1 - 2.97T + 67T^{2} \)
71 \( 1 + (5.97 + 5.97i)T + 71iT^{2} \)
73 \( 1 + (-11.8 - 11.8i)T + 73iT^{2} \)
79 \( 1 + (8.09 - 8.09i)T - 79iT^{2} \)
83 \( 1 + 5.07iT - 83T^{2} \)
89 \( 1 - 8.16T + 89T^{2} \)
97 \( 1 + (-5.16 - 5.16i)T + 97iT^{2} \)
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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

−11.17420242209977652226438458364, −10.46042471812034948817667960545, −9.599908661214685766340455896180, −8.341149325795085322106939068641, −7.78019026413272347754176913525, −7.08443874357816554384946871464, −5.70795990995273183425861186695, −4.74142250743265467706583059439, −3.73090942394239693437946802141, −2.05266271660104069332269535591, 1.43050939773521367587640350199, 2.52679844393801244346181426905, 3.27571125808881206486213684521, 5.06245547181340853119024272450, 6.05876625857836852627506870182, 7.56352408064781206025976111318, 8.228802298829311473723498467548, 8.951358682220416187696009733566, 10.32995437726502018856885192454, 11.01302443433801000365185691218

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