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Properties

Label 2-2e4-16.13-c1-0-0
Degree $2$
Conductor $16$
Sign $0.923 - 0.382i$
Analytic cond. $0.127760$
Root an. cond. $0.357436$
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 + i)2-s + (−1 − i)3-s − 2i·4-s + (−1 + i)5-s + 2·6-s + 2i·7-s + (2 + 2i)8-s − i·9-s − 2i·10-s + (1 − i)11-s + (−2 + 2i)12-s + (−1 − i)13-s + (−2 − 2i)14-s + 2·15-s − 4·16-s − 2·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.577 − 0.577i)3-s − i·4-s + (−0.447 + 0.447i)5-s + 0.816·6-s + 0.755i·7-s + (0.707 + 0.707i)8-s − 0.333i·9-s − 0.632i·10-s + (0.301 − 0.301i)11-s + (−0.577 + 0.577i)12-s + (−0.277 − 0.277i)13-s + (−0.534 − 0.534i)14-s + 0.516·15-s − 16-s − 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(0.127760\)
Root analytic conductor: \(0.357436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.359306 + 0.0714704i\)
\(L(\frac12)\) \(\approx\) \(0.359306 + 0.0714704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
5 \( 1 + (1 - i)T - 5iT^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (-3 - 3i)T + 29iT^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 + (3 - 3i)T - 59iT^{2} \)
61 \( 1 + (9 + 9i)T + 61iT^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (1 + i)T + 83iT^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−18.76920296060781340324975746860, −18.21324685159535169645627379191, −16.92264916008733610121958362645, −15.56248309308998716630007204974, −14.43163195458328134002338975151, −12.33488696765597057065259773085, −10.94615612885111038732160876916, −9.036487932412074199687555304026, −7.29041351208685469584643073213, −5.87627034365529106562658369692, 4.38815572194696855867145714346, 7.55228244623799316615009824381, 9.422303413938473441171880689948, 10.79350187367916155153671232394, 11.87534745135013107676801105826, 13.50004648340396989320504396892, 15.82802806170473633912930233477, 16.77693473848908654550406060602, 17.75343423278250220454729782949, 19.49325330313906922783187166826

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