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Properties

Label 2-104-104.69-c1-0-0
Degree $2$
Conductor $104$
Sign $0.869 - 0.494i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
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  + (−1.36 − 0.366i)2-s + (2.36 + 1.36i)3-s + (1.73 + i)4-s + 0.267·5-s + (−2.73 − 2.73i)6-s + (−3 + 1.73i)7-s + (−1.99 − 2i)8-s + (2.23 + 3.86i)9-s + (−0.366 − 0.0980i)10-s + (1 − 1.73i)11-s + (2.73 + 4.73i)12-s + (2.59 − 2.5i)13-s + (4.73 − 1.26i)14-s + (0.633 + 0.366i)15-s + (1.99 + 3.46i)16-s + (−3.23 − 5.59i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (1.36 + 0.788i)3-s + (0.866 + 0.5i)4-s + 0.119·5-s + (−1.11 − 1.11i)6-s + (−1.13 + 0.654i)7-s + (−0.707 − 0.707i)8-s + (0.744 + 1.28i)9-s + (−0.115 − 0.0310i)10-s + (0.301 − 0.522i)11-s + (0.788 + 1.36i)12-s + (0.720 − 0.693i)13-s + (1.26 − 0.338i)14-s + (0.163 + 0.0945i)15-s + (0.499 + 0.866i)16-s + (−0.783 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ 0.869 - 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.925430 + 0.244791i\)
\(L(\frac12)\) \(\approx\) \(0.925430 + 0.244791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
13 \( 1 + (-2.59 + 2.5i)T \)
good3 \( 1 + (-2.36 - 1.36i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.267T + 5T^{2} \)
7 \( 1 + (3 - 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.23 + 5.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.36 - 4.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.09 + 1.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.59 + 1.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + (-3.86 + 6.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.03 + 0.598i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.19 - 4.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.26iT - 47T^{2} \)
53 \( 1 - 9.92iT - 53T^{2} \)
59 \( 1 + (-3.73 - 6.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.36 + 9.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.0 - 6.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 5.46T + 83T^{2} \)
89 \( 1 + (0.464 + 0.267i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.91341122191977101398786655249, −12.90546026095736527275861276390, −11.53580243323461198907142928328, −10.21773925016823290779750898896, −9.387903522833939650182598234837, −8.834357663926325852606602758190, −7.70288829964671435338272721127, −6.09494337762618281372472474993, −3.60904537023455496062226159296, −2.67635887257639481241164146880, 1.84456934600409734569245529865, 3.52620897108498128919421212588, 6.46623293715454287649065243745, 7.10000603151769069608768436595, 8.312494517281831210401192353994, 9.210672170272203953931233207164, 10.03075847834763098474882288151, 11.50531432009185398468195912661, 13.05059792210574494949632383366, 13.59509263533686157123620621092

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