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Properties

Label 2-425-17.13-c1-0-18
Degree $2$
Conductor $425$
Sign $-0.0219 + 0.999i$
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  − 0.783i·2-s + (−0.385 − 0.385i)3-s + 1.38·4-s + (−0.301 + 0.301i)6-s + (0.840 − 0.840i)7-s − 2.65i·8-s − 2.70i·9-s + (−1.80 + 1.80i)11-s + (−0.534 − 0.534i)12-s + 0.368·13-s + (−0.658 − 0.658i)14-s + 0.693·16-s + (2.46 + 3.30i)17-s − 2.11·18-s − 6.61i·19-s + ⋯
L(s)  = 1  − 0.554i·2-s + (−0.222 − 0.222i)3-s + 0.693·4-s + (−0.123 + 0.123i)6-s + (0.317 − 0.317i)7-s − 0.937i·8-s − 0.901i·9-s + (−0.544 + 0.544i)11-s + (−0.154 − 0.154i)12-s + 0.102·13-s + (−0.175 − 0.175i)14-s + 0.173·16-s + (0.597 + 0.801i)17-s − 0.499·18-s − 1.51i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0219 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.0219 + 0.999i$
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: \((2,\ 425,\ (\ :1/2),\ -0.0219 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08555 - 1.10970i\)
\(L(\frac12)\) \(\approx\) \(1.08555 - 1.10970i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.46 - 3.30i)T \)
good2 \( 1 + 0.783iT - 2T^{2} \)
3 \( 1 + (0.385 + 0.385i)T + 3iT^{2} \)
7 \( 1 + (-0.840 + 0.840i)T - 7iT^{2} \)
11 \( 1 + (1.80 - 1.80i)T - 11iT^{2} \)
13 \( 1 - 0.368T + 13T^{2} \)
19 \( 1 + 6.61iT - 19T^{2} \)
23 \( 1 + (-2.73 + 2.73i)T - 23iT^{2} \)
29 \( 1 + (1.63 + 1.63i)T + 29iT^{2} \)
31 \( 1 + (-4.68 - 4.68i)T + 31iT^{2} \)
37 \( 1 + (2.24 + 2.24i)T + 37iT^{2} \)
41 \( 1 + (5.16 - 5.16i)T - 41iT^{2} \)
43 \( 1 + 6.82iT - 43T^{2} \)
47 \( 1 + 7.80T + 47T^{2} \)
53 \( 1 - 8.01iT - 53T^{2} \)
59 \( 1 - 5.22iT - 59T^{2} \)
61 \( 1 + (-5.74 + 5.74i)T - 61iT^{2} \)
67 \( 1 - 7.94T + 67T^{2} \)
71 \( 1 + (-8.40 - 8.40i)T + 71iT^{2} \)
73 \( 1 + (-10.4 - 10.4i)T + 73iT^{2} \)
79 \( 1 + (-0.575 + 0.575i)T - 79iT^{2} \)
83 \( 1 - 3.99iT - 83T^{2} \)
89 \( 1 - 9.14T + 89T^{2} \)
97 \( 1 + (-4.99 - 4.99i)T + 97iT^{2} \)
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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

−10.99137300005678875407177417234, −10.29032257932575103978848661925, −9.355572321864107902113411458310, −8.139397456595414237070999884767, −7.02922507265105095520130798789, −6.47253935729664095642078996556, −5.10841218424001226634570033760, −3.76479215707767962700575058585, −2.57896880192536085953765257936, −1.10099095083585080560776985323, 1.94812258682500143143470157713, 3.28986629576383943473126540518, 5.08962590047583952874909559659, 5.57995521522251503427073212933, 6.71279488658729974951335264898, 7.920973380947697486555660155803, 8.179500249958190327748720533387, 9.729398502388669890894842454513, 10.60493084873993249200663329453, 11.38927345044991279380759450901

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