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Properties

Label 2-425-17.13-c1-0-17
Degree $2$
Conductor $425$
Sign $-0.237 + 0.971i$
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  − 0.232i·2-s + (−1.69 − 1.69i)3-s + 1.94·4-s + (−0.393 + 0.393i)6-s + (1.46 − 1.46i)7-s − 0.917i·8-s + 2.73i·9-s + (−0.339 + 0.339i)11-s + (−3.29 − 3.29i)12-s + 4.07·13-s + (−0.339 − 0.339i)14-s + 3.67·16-s + (−1.69 − 3.75i)17-s + 0.635·18-s − 4i·19-s + ⋯
L(s)  = 1  − 0.164i·2-s + (−0.977 − 0.977i)3-s + 0.972·4-s + (−0.160 + 0.160i)6-s + (0.552 − 0.552i)7-s − 0.324i·8-s + 0.910i·9-s + (−0.102 + 0.102i)11-s + (−0.951 − 0.951i)12-s + 1.12·13-s + (−0.0907 − 0.0907i)14-s + 0.919·16-s + (−0.410 − 0.911i)17-s + 0.149·18-s − 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.237 + 0.971i$
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: \((2,\ 425,\ (\ :1/2),\ -0.237 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802086 - 1.02179i\)
\(L(\frac12)\) \(\approx\) \(0.802086 - 1.02179i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 3.75i)T \)
good2 \( 1 + 0.232iT - 2T^{2} \)
3 \( 1 + (1.69 + 1.69i)T + 3iT^{2} \)
7 \( 1 + (-1.46 + 1.46i)T - 7iT^{2} \)
11 \( 1 + (0.339 - 0.339i)T - 11iT^{2} \)
13 \( 1 - 4.07T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (5.76 - 5.76i)T - 23iT^{2} \)
29 \( 1 + (0.732 + 0.732i)T + 29iT^{2} \)
31 \( 1 + (4.28 + 4.28i)T + 31iT^{2} \)
37 \( 1 + (-0.917 - 0.917i)T + 37iT^{2} \)
41 \( 1 + (-7.62 + 7.62i)T - 41iT^{2} \)
43 \( 1 + 7.45iT - 43T^{2} \)
47 \( 1 - 3.60T + 47T^{2} \)
53 \( 1 - 6.14iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (4 - 4i)T - 61iT^{2} \)
67 \( 1 + 3.14T + 67T^{2} \)
71 \( 1 + (-1.28 - 1.28i)T + 71iT^{2} \)
73 \( 1 + (-8.60 - 8.60i)T + 73iT^{2} \)
79 \( 1 + (7.23 - 7.23i)T - 79iT^{2} \)
83 \( 1 - 2.23iT - 83T^{2} \)
89 \( 1 - 9.37T + 89T^{2} \)
97 \( 1 + (-11.8 - 11.8i)T + 97iT^{2} \)
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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

−11.23966951564976349498906113893, −10.47964968784816312703609986654, −9.118323897474771396729760944870, −7.60765738659640820245439029484, −7.27562301236087894474176430267, −6.21995620767183449296280769303, −5.50697518054177723921838070692, −3.91905086871856876772847291992, −2.20471432363156871625164236233, −0.983179768358885066472421373398, 1.89220440696529597323553311509, 3.62208818631898755271731035891, 4.78091892699501123480462057498, 6.03037948177202168929483933623, 6.17379674605973367786725119080, 7.84649546921267388535406417350, 8.623218250293102675575474382135, 9.990324841092045557787515993183, 10.77253310373154961513524145683, 11.18433065918466163742428152268

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