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Properties

Label 2-476-7.5-c2-0-10
Degree $2$
Conductor $476$
Sign $0.385 - 0.922i$
Analytic cond. $12.9700$
Root an. cond. $3.60139$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.113 + 0.0652i)3-s + (6.14 + 3.54i)5-s + (−1.76 − 6.77i)7-s + (−4.49 + 7.77i)9-s + (5.40 + 9.35i)11-s + 11.8i·13-s − 0.925·15-s + (3.57 − 2.06i)17-s + (−1.70 − 0.986i)19-s + (0.641 + 0.650i)21-s + (−1.14 + 1.98i)23-s + (12.6 + 21.8i)25-s − 2.34i·27-s + 35.4·29-s + (−15.6 + 9.04i)31-s + ⋯
L(s)  = 1  + (−0.0376 + 0.0217i)3-s + (1.22 + 0.709i)5-s + (−0.252 − 0.967i)7-s + (−0.499 + 0.864i)9-s + (0.490 + 0.850i)11-s + 0.915i·13-s − 0.0617·15-s + (0.210 − 0.121i)17-s + (−0.0899 − 0.0519i)19-s + (0.0305 + 0.0309i)21-s + (−0.0498 + 0.0863i)23-s + (0.505 + 0.875i)25-s − 0.0869i·27-s + 1.22·29-s + (−0.505 + 0.291i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.385 - 0.922i$
Analytic conductor: \(12.9700\)
Root analytic conductor: \(3.60139\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1),\ 0.385 - 0.922i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.936195225\)
\(L(\frac12)\) \(\approx\) \(1.936195225\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.76 + 6.77i)T \)
17 \( 1 + (-3.57 + 2.06i)T \)
good3 \( 1 + (0.113 - 0.0652i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-6.14 - 3.54i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-5.40 - 9.35i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 11.8iT - 169T^{2} \)
19 \( 1 + (1.70 + 0.986i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (1.14 - 1.98i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 35.4T + 841T^{2} \)
31 \( 1 + (15.6 - 9.04i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (15.7 - 27.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 42.5iT - 1.68e3T^{2} \)
43 \( 1 - 25.7T + 1.84e3T^{2} \)
47 \( 1 + (14.6 + 8.46i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-3.79 - 6.57i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-54.6 + 31.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-77.5 - 44.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (5.04 + 8.74i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 127.T + 5.04e3T^{2} \)
73 \( 1 + (8.61 - 4.97i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (46.3 - 80.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 6.00iT - 6.88e3T^{2} \)
89 \( 1 + (1.67 + 0.966i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 19.3iT - 9.40e3T^{2} \)
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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

−10.79070459518223394144927533986, −10.04442356926023148691262595631, −9.502133245591439367414166793727, −8.232801166342483481187383546826, −6.98408544532547117125480871548, −6.55841916667938579753279203552, −5.31206505694770677974404255158, −4.21863130691152859950181163078, −2.74686954098380773349424302888, −1.62466467336491369095394392018, 0.823343849438311094368358830568, 2.37522179307051626949097041472, 3.56357997204748459555343909378, 5.31542985939731857393997715823, 5.80120545046217030315634369205, 6.57441974677831915276396485272, 8.274053142462931105049460168102, 8.935952020786762893538098166397, 9.527769140775750234893559672774, 10.50417920046945419502775035364

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