Properties

Label 2-154-77.76-c1-0-2
Degree $2$
Conductor $154$
Sign $-0.818 - 0.575i$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
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  + i·2-s + 3.09i·3-s − 4-s + 0.646i·5-s − 3.09·6-s + (2.44 − i)7-s i·8-s − 6.58·9-s − 0.646·10-s + (2.79 − 1.79i)11-s − 3.09i·12-s − 3.09·13-s + (1 + 2.44i)14-s − 2·15-s + 16-s + 3.74·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.78i·3-s − 0.5·4-s + 0.288i·5-s − 1.26·6-s + (0.925 − 0.377i)7-s − 0.353i·8-s − 2.19·9-s − 0.204·10-s + (0.841 − 0.540i)11-s − 0.893i·12-s − 0.858·13-s + (0.267 + 0.654i)14-s − 0.516·15-s + 0.250·16-s + 0.907·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.818 - 0.575i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ -0.818 - 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.335621 + 1.06114i\)
\(L(\frac12)\) \(\approx\) \(0.335621 + 1.06114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 + (-2.44 + i)T \)
11 \( 1 + (-2.79 + 1.79i)T \)
good3 \( 1 - 3.09iT - 3T^{2} \)
5 \( 1 - 0.646iT - 5T^{2} \)
13 \( 1 + 3.09T + 13T^{2} \)
17 \( 1 - 3.74T + 17T^{2} \)
19 \( 1 + 5.54T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 7.58iT - 29T^{2} \)
31 \( 1 - 1.15iT - 31T^{2} \)
37 \( 1 + 5.58T + 37T^{2} \)
41 \( 1 - 5.03T + 41T^{2} \)
43 \( 1 + 11.1iT - 43T^{2} \)
47 \( 1 + 5.03iT - 47T^{2} \)
53 \( 1 + 2.41T + 53T^{2} \)
59 \( 1 + 3.09iT - 59T^{2} \)
61 \( 1 + 9.28T + 61T^{2} \)
67 \( 1 - 1.58T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 6.32T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 9.15T + 83T^{2} \)
89 \( 1 + 9.79iT - 89T^{2} \)
97 \( 1 - 15.9iT - 97T^{2} \)
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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

−14.01083944654777920099804614581, −12.21959624308603316056028426694, −10.92088157760920941957227549567, −10.41485365587145361610647315165, −9.164506638359341977793587903901, −8.477859929236906606030906565214, −6.98532454069619333608950927538, −5.43287976001281930911638654443, −4.60205458637534246654963088967, −3.45820939089626915951412972212, 1.33009268067362763655028261273, 2.50826445857973340822604385999, 4.72325137684521762672321986146, 6.15963905710072921472390323943, 7.42653310206753283958492086757, 8.285962721378133643722003930616, 9.323731687783283821030265646209, 10.95319850514513080677873629050, 11.95861794516422487475120467192, 12.38647099815295859349060340788

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