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Properties

Label 2-425-17.16-c1-0-2
Degree $2$
Conductor $425$
Sign $-0.765 + 0.643i$
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  − 1.48·2-s + 3.29i·3-s + 0.193·4-s − 4.87i·6-s + 2.22i·7-s + 2.67·8-s − 7.83·9-s + 2.86i·11-s + 0.638i·12-s − 2.28·13-s − 3.29i·14-s − 4.35·16-s + (−3.15 + 2.65i)17-s + 11.5·18-s + 5.76·19-s + ⋯
L(s)  = 1  − 1.04·2-s + 1.90i·3-s + 0.0969·4-s − 1.99i·6-s + 0.839i·7-s + 0.945·8-s − 2.61·9-s + 0.862i·11-s + 0.184i·12-s − 0.634·13-s − 0.879i·14-s − 1.08·16-s + (−0.765 + 0.643i)17-s + 2.73·18-s + 1.32·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.765 + 0.643i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.765 + 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153452 - 0.421073i\)
\(L(\frac12)\) \(\approx\) \(0.153452 - 0.421073i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.15 - 2.65i)T \)
good2 \( 1 + 1.48T + 2T^{2} \)
3 \( 1 - 3.29iT - 3T^{2} \)
7 \( 1 - 2.22iT - 7T^{2} \)
11 \( 1 - 2.86iT - 11T^{2} \)
13 \( 1 + 2.28T + 13T^{2} \)
19 \( 1 - 5.76T + 19T^{2} \)
23 \( 1 - 1.58iT - 23T^{2} \)
29 \( 1 + 9.23iT - 29T^{2} \)
31 \( 1 - 1.15iT - 31T^{2} \)
37 \( 1 - 0.514iT - 37T^{2} \)
41 \( 1 + 7.09iT - 41T^{2} \)
43 \( 1 - 7.89T + 43T^{2} \)
47 \( 1 + 3.03T + 47T^{2} \)
53 \( 1 + 5.73T + 53T^{2} \)
59 \( 1 + 7.50T + 59T^{2} \)
61 \( 1 - 11.8iT - 61T^{2} \)
67 \( 1 + 7.35T + 67T^{2} \)
71 \( 1 - 8.80iT - 71T^{2} \)
73 \( 1 + 2.65iT - 73T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + 3.08T + 83T^{2} \)
89 \( 1 - 2.15T + 89T^{2} \)
97 \( 1 - 8.72iT - 97T^{2} \)
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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

−11.32503610489137677821829991784, −10.43788276587591159025994651766, −9.725760904200699946715701566231, −9.299237201096574965451384406429, −8.544647495314160346399059923491, −7.50494492577960229980214153999, −5.79379866908719785234998721081, −4.84279575099828235568239887195, −4.03844607275980575191854673271, −2.47417221618107031232921238296, 0.42447710879453842163520309327, 1.47921618965667937029441599048, 3.00616398430040188443703104922, 4.98372966077137501091218802005, 6.36535364271805055290821939410, 7.31959217114707118294159658319, 7.66225874173090081119555413256, 8.656070739491323047218222567413, 9.429018668415182756661130194561, 10.77111663668707975211455363673

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