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Properties

Label 2-416-32.29-c1-0-13
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: $\chi_{416} (157, \cdot )$
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

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

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