Properties

Label 1-68-68.7-r0-0-0
Degree $1$
Conductor $68$
Sign $0.226 + 0.974i$
Analytic cond. $0.315790$
Root an. cond. $0.315790$
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  + (0.382 + 0.923i)3-s + (−0.923 + 0.382i)5-s + (0.923 + 0.382i)7-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s i·13-s + (−0.707 − 0.707i)15-s + (0.707 + 0.707i)19-s + i·21-s + (0.382 − 0.923i)23-s + (0.707 − 0.707i)25-s + (−0.923 − 0.382i)27-s + (0.923 − 0.382i)29-s + (−0.382 − 0.923i)31-s − 33-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (−0.923 + 0.382i)5-s + (0.923 + 0.382i)7-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s i·13-s + (−0.707 − 0.707i)15-s + (0.707 + 0.707i)19-s + i·21-s + (0.382 − 0.923i)23-s + (0.707 − 0.707i)25-s + (−0.923 − 0.382i)27-s + (0.923 − 0.382i)29-s + (−0.382 − 0.923i)31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.226 + 0.974i$
Analytic conductor: \(0.315790\)
Root analytic conductor: \(0.315790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 68,\ (0:\ ),\ 0.226 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7355505736 + 0.5843222257i\)
\(L(\frac12)\) \(\approx\) \(0.7355505736 + 0.5843222257i\)
\(L(1)\) \(\approx\) \(0.9443061744 + 0.4199308384i\)
\(L(1)\) \(\approx\) \(0.9443061744 + 0.4199308384i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (0.923 + 0.382i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 - iT \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.923 - 0.382i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (-0.382 - 0.923i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.923 + 0.382i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.923 - 0.382i)T \)
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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

−31.26117005606192464087187970229, −30.92246298035136698804529331195, −29.62539711742217294342152997618, −28.5703123076298013712554958251, −27.17689547841740978044770460576, −26.34974864567622398252931512839, −24.818716774701306028602341215764, −23.84731564327044535122866859586, −23.4883878800814163254673331072, −21.52811258592806533864452498909, −20.31310425919756988637207065901, −19.38866578189246723983743720323, −18.380364191045791780462837355348, −17.10969136752935463274076412824, −15.761290852140284640287931128133, −14.32716987304950328690157212886, −13.40874219607682440901978283141, −11.93361912440944548204617045071, −11.16765827970942081584079134056, −8.90622038064882185642092124723, −7.97844637478264944047698267362, −6.91410280533428067886240571121, −4.981361421091537974968057854534, −3.29824360252817815506395422952, −1.2939612276670734417592203602, 2.63852168854888205208940764878, 4.135073604262902445775594756, 5.31532210919947948252396777270, 7.571936324624049233450331493534, 8.467796374136064714519430362318, 10.094658899700320153740435448953, 11.11186057373821780131180580609, 12.382005586334026035158121385398, 14.31484484114913413990603565786, 15.13971080528547196703241369671, 15.924617274045452965092161242769, 17.516893117785268874948941832346, 18.769112160384508074344542876490, 20.20708289276609101304511490603, 20.810404247939170302704655399782, 22.28427618520297255186074601274, 23.07562638261696675718215502289, 24.58025096399572127350054250903, 25.73527793401785434806002669658, 26.97137937013894714436703981030, 27.53219188844512795168478951977, 28.551113698918095228051001936308, 30.49461669828625147117906654616, 31.03181185753534237315346196962, 32.05407155458239634877225146027

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