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Properties

Label 2-416-32.21-c1-0-38
Degree $2$
Conductor $416$
Sign $0.900 + 0.434i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
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  + (−0.705 + 1.22i)2-s + (3.06 − 1.26i)3-s + (−1.00 − 1.72i)4-s + (0.112 − 0.272i)5-s + (−0.604 + 4.65i)6-s + (−1.80 − 1.80i)7-s + (2.82 − 0.0142i)8-s + (5.66 − 5.66i)9-s + (0.254 + 0.330i)10-s + (−2.10 − 0.872i)11-s + (−5.27 − 4.02i)12-s + (−0.382 − 0.923i)13-s + (3.49 − 0.942i)14-s − 0.978i·15-s + (−1.97 + 3.47i)16-s − 3.28i·17-s + ⋯
L(s)  = 1  + (−0.498 + 0.866i)2-s + (1.76 − 0.733i)3-s + (−0.502 − 0.864i)4-s + (0.0504 − 0.121i)5-s + (−0.246 + 1.89i)6-s + (−0.683 − 0.683i)7-s + (0.999 − 0.00503i)8-s + (1.88 − 1.88i)9-s + (0.0804 + 0.104i)10-s + (−0.635 − 0.263i)11-s + (−1.52 − 1.16i)12-s + (−0.106 − 0.256i)13-s + (0.933 − 0.251i)14-s − 0.252i·15-s + (−0.494 + 0.869i)16-s − 0.796i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66480 - 0.380827i\)
\(L(\frac12)\) \(\approx\) \(1.66480 - 0.380827i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.705 - 1.22i)T \)
13 \( 1 + (0.382 + 0.923i)T \)
good3 \( 1 + (-3.06 + 1.26i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.112 + 0.272i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.80 + 1.80i)T + 7iT^{2} \)
11 \( 1 + (2.10 + 0.872i)T + (7.77 + 7.77i)T^{2} \)
17 \( 1 + 3.28iT - 17T^{2} \)
19 \( 1 + (-2.88 - 6.96i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.45 + 1.45i)T - 23iT^{2} \)
29 \( 1 + (7.19 - 2.98i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.16T + 31T^{2} \)
37 \( 1 + (-1.93 + 4.67i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.89 - 1.89i)T - 41iT^{2} \)
43 \( 1 + (-9.69 - 4.01i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 10.8iT - 47T^{2} \)
53 \( 1 + (-6.82 - 2.82i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.41 - 3.42i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.10 - 1.28i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (2.33 - 0.966i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-3.12 - 3.12i)T + 71iT^{2} \)
73 \( 1 + (-1.58 + 1.58i)T - 73iT^{2} \)
79 \( 1 - 15.9iT - 79T^{2} \)
83 \( 1 + (-1.02 - 2.47i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (5.24 + 5.24i)T + 89iT^{2} \)
97 \( 1 + 5.34T + 97T^{2} \)
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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

−10.73257543689573844876353444278, −9.680858746276130612484275115851, −9.254932180065360623497142983438, −8.197664334989477962107673188815, −7.52201728360333047792761597069, −6.98928393763668145356291748406, −5.69038402301926331667775798826, −4.02501046614192924067452124004, −2.87790544769881024505718925353, −1.20483894783509966937687639681, 2.21817099434297888204050207626, 2.88901046458464814499449274698, 3.87867102541299790008120591028, 5.00450393358026652084033995084, 7.09731868709974497543551539918, 8.002790124869597470822274973405, 8.934473687152927753033616971524, 9.332377314654938639370866310934, 10.14362340800335090968849989533, 10.90912200384046963151457890936

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