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Properties

Label 1-668-668.611-r0-0-0
Degree $1$
Conductor $668$
Sign $-0.698 + 0.715i$
Analytic cond. $3.10217$
Root an. cond. $3.10217$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.898 − 0.438i)3-s + (−0.862 + 0.505i)5-s + (−0.954 − 0.298i)7-s + (0.614 + 0.788i)9-s + (0.644 + 0.764i)11-s + (0.387 − 0.922i)13-s + (0.997 − 0.0756i)15-s + (0.982 + 0.188i)17-s + (−0.776 − 0.629i)19-s + (0.726 + 0.686i)21-s + (−0.584 − 0.811i)23-s + (0.489 − 0.872i)25-s + (−0.206 − 0.978i)27-s + (−0.584 + 0.811i)29-s + (0.843 − 0.537i)31-s + ⋯
L(s)  = 1  + (−0.898 − 0.438i)3-s + (−0.862 + 0.505i)5-s + (−0.954 − 0.298i)7-s + (0.614 + 0.788i)9-s + (0.644 + 0.764i)11-s + (0.387 − 0.922i)13-s + (0.997 − 0.0756i)15-s + (0.982 + 0.188i)17-s + (−0.776 − 0.629i)19-s + (0.726 + 0.686i)21-s + (−0.584 − 0.811i)23-s + (0.489 − 0.872i)25-s + (−0.206 − 0.978i)27-s + (−0.584 + 0.811i)29-s + (0.843 − 0.537i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $-0.698 + 0.715i$
Analytic conductor: \(3.10217\)
Root analytic conductor: \(3.10217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{668} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 668,\ (0:\ ),\ -0.698 + 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07898217491 + 0.1874009230i\)
\(L(\frac12)\) \(\approx\) \(0.07898217491 + 0.1874009230i\)
\(L(1)\) \(\approx\) \(0.5484072273 + 0.004327841038i\)
\(L(1)\) \(\approx\) \(0.5484072273 + 0.004327841038i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (-0.898 - 0.438i)T \)
5 \( 1 + (-0.862 + 0.505i)T \)
7 \( 1 + (-0.954 - 0.298i)T \)
11 \( 1 + (0.644 + 0.764i)T \)
13 \( 1 + (0.387 - 0.922i)T \)
17 \( 1 + (0.982 + 0.188i)T \)
19 \( 1 + (-0.776 - 0.629i)T \)
23 \( 1 + (-0.584 - 0.811i)T \)
29 \( 1 + (-0.584 + 0.811i)T \)
31 \( 1 + (0.843 - 0.537i)T \)
37 \( 1 + (-0.614 + 0.788i)T \)
41 \( 1 + (-0.421 + 0.906i)T \)
43 \( 1 + (-0.521 - 0.853i)T \)
47 \( 1 + (-0.672 + 0.739i)T \)
53 \( 1 + (-0.351 + 0.936i)T \)
59 \( 1 + (-0.982 + 0.188i)T \)
61 \( 1 + (-0.914 + 0.404i)T \)
67 \( 1 + (0.862 + 0.505i)T \)
71 \( 1 + (0.280 + 0.959i)T \)
73 \( 1 + (0.999 - 0.0378i)T \)
79 \( 1 + (-0.993 - 0.113i)T \)
83 \( 1 + (-0.700 + 0.713i)T \)
89 \( 1 + (0.997 + 0.0756i)T \)
97 \( 1 + (-0.843 - 0.537i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.81932961075383396001185249827, −21.49284675751889285547211663220, −21.21797004747307625601571136973, −19.92634242273831516819082973946, −19.0687684319430220686447039344, −18.64569553008267660252466039413, −17.19395151853207889119394612336, −16.589089946140874794484996809873, −16.05236843562632326302900614245, −15.36634558481426990866674944265, −14.22014783174959444604536043799, −13.09626669677950722116382297299, −12.05833437296910946813750526576, −11.81551542993448383772571318241, −10.78753677637881434279007323480, −9.71831497333043904327815213159, −9.03246901197567167337454655887, −7.981359233107279300169000968658, −6.71075311625677789190001176732, −6.060402226664650512054691239095, −5.08659049177823016868453498138, −3.84712294303515233951566258332, −3.528326009072557693351647614787, −1.5031756767222714451527547749, −0.13023785300319758206436464577, 1.20913463112536703098359154228, 2.7786396723171038845524891977, 3.82232054876455989317214458125, 4.7530064781301092005505599231, 6.07971669790333838936166146582, 6.69315432067922839962673658596, 7.47339160190765666197153712159, 8.39445366620902532251237680385, 9.89687905303999919895291948395, 10.484832623672227989504098357510, 11.37383914054990549464277511713, 12.35797639237506741548620519314, 12.71219214148153931306708594987, 13.84554901784029462597605784294, 15.02606613337674421005480598099, 15.67369239084549144124301549248, 16.63865357648386896933506848886, 17.21753391151617600681368348415, 18.31615250603949185905414995349, 18.890036986002163542924245405569, 19.72381887330002491495939461770, 20.39532902500990556459435400488, 21.85152696912212680850642051654, 22.520512905257113741845865481147, 23.0364333596581181876150163420

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