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Properties

Label 2-26e2-52.19-c1-0-48
Degree $2$
Conductor $676$
Sign $0.909 - 0.416i$
Analytic cond. $5.39788$
Root an. cond. $2.32333$
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  + (1.39 + 0.236i)2-s + (0.736 + 0.425i)3-s + (1.88 + 0.659i)4-s + (0.166 − 0.166i)5-s + (0.925 + 0.766i)6-s + (2.55 + 0.684i)7-s + (2.47 + 1.36i)8-s + (−1.13 − 1.97i)9-s + (0.271 − 0.192i)10-s + (−0.373 − 1.39i)11-s + (1.10 + 1.28i)12-s + (3.39 + 1.55i)14-s + (0.193 − 0.0517i)15-s + (3.12 + 2.49i)16-s + (−1.21 + 0.699i)17-s + (−1.12 − 3.01i)18-s + ⋯
L(s)  = 1  + (0.985 + 0.167i)2-s + (0.425 + 0.245i)3-s + (0.944 + 0.329i)4-s + (0.0744 − 0.0744i)5-s + (0.377 + 0.313i)6-s + (0.965 + 0.258i)7-s + (0.875 + 0.483i)8-s + (−0.379 − 0.657i)9-s + (0.0858 − 0.0609i)10-s + (−0.112 − 0.419i)11-s + (0.320 + 0.371i)12-s + (0.908 + 0.416i)14-s + (0.0498 − 0.0133i)15-s + (0.782 + 0.622i)16-s + (−0.293 + 0.169i)17-s + (−0.264 − 0.711i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.909 - 0.416i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ 0.909 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.29683 + 0.719564i\)
\(L(\frac12)\) \(\approx\) \(3.29683 + 0.719564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.236i)T \)
13 \( 1 \)
good3 \( 1 + (-0.736 - 0.425i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.166 + 0.166i)T - 5iT^{2} \)
7 \( 1 + (-2.55 - 0.684i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.373 + 1.39i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.21 - 0.699i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.44 + 5.39i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.37 - 7.57i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.11 - 3.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.88 + 3.88i)T + 31iT^{2} \)
37 \( 1 + (0.5 - 0.133i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.5 - 5.59i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.59 + 7.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.80 + 2.80i)T - 47iT^{2} \)
53 \( 1 + 5.94T + 53T^{2} \)
59 \( 1 + (-8.22 - 2.20i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.61 - 6.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.43 - 0.652i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.81 + 10.5i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.05 + 5.05i)T + 73iT^{2} \)
79 \( 1 + 8.51iT - 79T^{2} \)
83 \( 1 + (-6.91 - 6.91i)T + 83iT^{2} \)
89 \( 1 + (6.41 - 1.71i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (14.9 + 4.00i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.94427277695673174265752542294, −9.590100704211301359179456343052, −8.747512471131300850904410144021, −7.86071103869666577510365584106, −6.98138237938195602369461233999, −5.77428568884770904830842735984, −5.16230456876212554147980725236, −3.97531859795753117829315045221, −3.10848148456093301026064556665, −1.81355996278596593184140009254, 1.74736493729869833488248614778, 2.58398838740609663913464126394, 3.99499874802030841039042465692, 4.83413073039243328405900315485, 5.77399829496812718587299766203, 6.83946852905585591554814494462, 7.86417636085381075799520198802, 8.321593860765903540948763941091, 9.869749949218228207787434332671, 10.63994768483640394468492271064

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