Properties

Label 2-116-116.75-c1-0-10
Degree $2$
Conductor $116$
Sign $-0.128 + 0.991i$
Analytic cond. $0.926264$
Root an. cond. $0.962426$
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  + (−0.431 − 1.34i)2-s + (0.427 + 0.427i)3-s + (−1.62 + 1.16i)4-s − 1.96i·5-s + (0.391 − 0.761i)6-s − 4.01i·7-s + (2.26 + 1.68i)8-s − 2.63i·9-s + (−2.65 + 0.850i)10-s + (3.24 + 3.24i)11-s + (−1.19 − 0.198i)12-s + 0.829i·13-s + (−5.40 + 1.73i)14-s + (0.842 − 0.842i)15-s + (1.29 − 3.78i)16-s + (−5.68 + 5.68i)17-s + ⋯
L(s)  = 1  + (−0.305 − 0.952i)2-s + (0.247 + 0.247i)3-s + (−0.813 + 0.581i)4-s − 0.880i·5-s + (0.159 − 0.310i)6-s − 1.51i·7-s + (0.802 + 0.596i)8-s − 0.877i·9-s + (−0.838 + 0.269i)10-s + (0.977 + 0.977i)11-s + (−0.344 − 0.0572i)12-s + 0.230i·13-s + (−1.44 + 0.462i)14-s + (0.217 − 0.217i)15-s + (0.323 − 0.946i)16-s + (−1.37 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $-0.128 + 0.991i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :1/2),\ -0.128 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.607320 - 0.690939i\)
\(L(\frac12)\) \(\approx\) \(0.607320 - 0.690939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.431 + 1.34i)T \)
29 \( 1 + (-2.85 + 4.56i)T \)
good3 \( 1 + (-0.427 - 0.427i)T + 3iT^{2} \)
5 \( 1 + 1.96iT - 5T^{2} \)
7 \( 1 + 4.01iT - 7T^{2} \)
11 \( 1 + (-3.24 - 3.24i)T + 11iT^{2} \)
13 \( 1 - 0.829iT - 13T^{2} \)
17 \( 1 + (5.68 - 5.68i)T - 17iT^{2} \)
19 \( 1 + (-1.50 - 1.50i)T + 19iT^{2} \)
23 \( 1 - 4.86iT - 23T^{2} \)
31 \( 1 + (-0.354 - 0.354i)T + 31iT^{2} \)
37 \( 1 + (-0.708 - 0.708i)T + 37iT^{2} \)
41 \( 1 + (-5.49 - 5.49i)T + 41iT^{2} \)
43 \( 1 + (4.40 + 4.40i)T + 43iT^{2} \)
47 \( 1 + (3.76 - 3.76i)T - 47iT^{2} \)
53 \( 1 - 3.48T + 53T^{2} \)
59 \( 1 + 6.43iT - 59T^{2} \)
61 \( 1 + (-1.74 + 1.74i)T - 61iT^{2} \)
67 \( 1 + 1.78T + 67T^{2} \)
71 \( 1 + 4.74T + 71T^{2} \)
73 \( 1 + (1.66 + 1.66i)T + 73iT^{2} \)
79 \( 1 + (-7.35 - 7.35i)T + 79iT^{2} \)
83 \( 1 - 14.3iT - 83T^{2} \)
89 \( 1 + (7.39 - 7.39i)T - 89iT^{2} \)
97 \( 1 + (3.56 + 3.56i)T + 97iT^{2} \)
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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

−13.08763080276436739262446495312, −12.24886046591000908465615690282, −11.17248543577177478605289891207, −9.970778661083132542559589750474, −9.298926043328830720598019472333, −8.198000863216242984793460684749, −6.77040257446188712584748111970, −4.42752089128659871240910141176, −3.87438202838753896535804630993, −1.35311459594862563792131884700, 2.69244280397057102281288163492, 4.96253501991233551203753010192, 6.23266279688617366468602048382, 7.13732414551155207655712311664, 8.581201186546619054577197277807, 9.084563368611072898661452697810, 10.64704650196423536084248163393, 11.65775256982918314893123643471, 13.17027728075037425243954691034, 14.11738149202408370261181470002

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